hopfield neural network method for problem of telescoping

Research ArticleHopfield Neural Network Method for Problem ofTelescoping Path Optimization of Single-Cylinder Pin-TypeMultisection Boom

YanMao and Kai Cheng

School of Mechanical and Aerospace Engineering Jilin University Changchun 130025 China

Correspondence should be addressed to Kai Cheng chengkaijlueducn

Received 6 April 2019 Accepted 13 June 2019 Published 8 July 2019

Academic Editor Alessandro Formisano

Copyright copy 2019 Yan Mao and Kai Cheng This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Telescoping path optimization (TPO) of single-cylinder pin-type multisection boom (SPMB) is a practical engineering problemthat is valuable to investigate This article studies the TPO problem and finds the key of TPO is to obtain the maximum retractionbackmost combination Amathematic model on the basis of the quadratic penalty function of a Hopfield neural network (HNN) isconstructed Two strategies are presented to improve the performance of TPOmodel one is proportional integral derivative (PID)strategy that adaptively adjusts the parameter 120582 of the constrained term and the parameter 120574 of the optimization objective term bycontrolling the value of constraint violation 119892119896 and the other is efficiency factor strategy that an efficiency factor is introduced inmodel for prioritizing the constrained term over the objective term Data test shows that compared with the path of boom lengthchanging before optimization both the number of sections that need to be moved and the total travels of cylinder can be reducedby 10-30 after optimization Both the PID strategy and the efficiency factor strategy achieve good optimization effects Theefficiency factor strategy is excellent at moderating the conflicts between the constrained term and the objective term thus thegenerations of the valid and the optimal solutions get well improved

1 Introduction

The upstructure of a mobile crane is composed of fourmajor mechanisms slewer derricking mechanism winchand telescopic boom Single-cylinder pin-type telescopicmultisection boom (SPMB) is the structure of telescopicboom The SPMB has a fixed section and multiple telescopicsections and the telescopic sections are sleeved one by onein the way that the small section is inserted in the largeone The length of each section is divided into several scalesand each scale has a hole set for being locked by pin of theinner adjacent section Therefore the different hole of outersection being locked by the pin of inner section determinesthe different stretching of the inner section A single longcylinder driving boom sections sequentially and the sectionsare pined gradually to keep the extension of the boom Boomlength is determined by all sectionsrsquo stretching and eachsectionrsquos stretching is decided by the location of the hole being

pinned of its outer section Obviously the boom length valuesof SPMB are discontinuous

Single-cylinder pin-type multisection boom (SPMB) hasgood load bearing performance that is mainly used in largetons automobile crane at present For example the truckcranes with over 100 tons lift weight are most equippedwith SPMB mechanisms a giant all terrain crane with liftweight of 2000 tons has an eight-section SPMB whose boomlength exceeds 100 meters when the SPMB is being fullystretched For now the material of SPMB is steel for craneand the maximum number of boom sections is only eightowing to weight limit of material In future with the designoptimization and the applications of light-weight materialssuch as carbon fiber and polymer materials the sectionsnumber might be manufactured more than the eight mightbe n=102030 and so on

However the telescoping efficiency of SPMB is lowBecause all the sections are driven by a long single cylinder

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 6195013 14 pageshttpsdoiorg10115520196195013

2 Mathematical Problems in Engineering

the multiple sections must await to be retracted one by one insequence from initial position to backmost position and thenwait to be stretched in reverse sequence from the backmostposition to target position while changing boom length fromone state to another When telescoping one section a setof complex procedure (telescoping step) must be operatedmainly involving processes of cylinder freely moving towardsaimed section and cylinder driving the aimed section towardstarget position Provided that a boom has 119899 sections (2n-1) telescoping steps should be operated when changing theboom length where (n-1) steps are for retracting sectionsand 119899 steps are for stretching sections If optimization isperformed the telescoping steps can be reduced by one ortwo and then the work efficiency can be evidently improved

TPO of SPMB is a practical engineering problemAlthough SPMB is widely applied in machinery and equip-ment research on TPO is lacking

Reference [1] (2012) described the TPO problem firstin public and optimized a five-section SPMB in a nestedprogram The principle was that judging the sections fromthe outermost section to the innermost section if therequired cylinder travel exceeded the offered cylinder travelretracted the section fully otherwise it kept the sectionunmoved [1] Reference [1] is enumeration method andmakes judgment only between two states of ldquoretract fullyrdquoand ldquokeep unmovedrdquo Thus the effect of optimization is verylimited

Mao Y et al (2018) proposed a simplified Permutationand Combination (PampC) method that chose three states foreach section to participate PampC those were ldquohold at initialpositionrdquo ldquomove to target positionrdquo and ldquoretract to full backrdquo(actually there are usually more states than these three Ifthere are number119898 holes on a section there will be number119898 stretched states for the section) Took these three states ofall sections to form state combinations and to participate PampCtogether then eliminate the state combinations exceedingcylinder travel allowance (invalid solutions) by function eval-uations and finally pick out the state combinations havingthe shortest paths (optimal solutions) from the remainingstate combinations (valid solutions) [2] PampC algorithmbelongs to enumeration category which can get optimalsolutions in short time when boom sections are few But ifthe number of boom sections increases its calculation willgrow up exponentially Besides the simplified handling thatonly three states for each section are chosen to form statecombinations of all sections cannot describe all telescopingpath possibilities

Reference [3] (2018) is another method to calculatethe TPO The method evaluated permitted cylinder travelallowance starting from the endmost section back to theforemost section Through two function evaluations of max-imum permitted travel allowance and minimum permittedtravel allowance it was decided whether opening the sidefunction evaluations or retracting one section full back andthen continuing downward evaluations [3]

References [1ndash3] are enumeration methods in nature andare suitable for small-scale problems because their logic willbe increasingly complicated with problem scales enlargingBesides thesemethods only pick out limited number of states

of the holes locations to make judgments in order to simplifycalculations and cannot reflect all the path possibilities sotheir optimization effects are not very well

TPO refers to the fact that when telescoping a multisec-tion boom from an initial state to a target state the numberof telescoping steps is the smallest and the total travelsof the driving cylinder is the shortest TPO is essentiallythe shortest-path scheduling problem of combination opti-mization problem (COP) TPO aims to work out maximumretraction backmost combination (RBC) that is the RBCposition always exists while all sections retract from theinitial position combination After retracting sections to theRBC position sections should be stretched to target positioncombination subsequently RBC length is limited in thefull travel of the long cylinder that is set as ldquo1rdquo The TPOproblem has the following features Movements of sectionsshould be operated in sequence When one section is readyto be retracted or stretched the current sufficiency of thetravel allowance of the driving cylinder should be consideredMoreover one sectionrsquos movement can affect other sectionsrsquomovability Thus TPO contains multiple constraints thatassociate with each other

Small-scale COPs can be solved by exact algorithms suchas dynamic programming branch definition and enumer-ation Heuristics can quickly calculate approximate optimalsolutions for large-scale complex problems but their opti-mality cannot be guaranteed Heuristics include neighboralgorithms simulated annealing evolutionary algorithmsand neural networks

The Hopfield neural network (HNN) has advantages insolving COPs In practical applications when converting theobjective function of the optimization problem to the energyequation of HNN to map variables to neural states in thenetwork HNN can be used to solve COPs That is when theneural state of the network tends to equilibrium the energyequation of the network converges to minimum the net-workrsquos convergence from initial state to steady state illustratesthe optimization calculating process of the objective function[4]

The energy equation is convenient to constraints dis-posing yet HNN has following difficulties (I) the penaltyparameters of energy equation are difficult to determine(II) the convergence is often trapped in local minimum(III) for problems of the optimal solutions distributing onboundary of constraints the adjustments of constrained termsometimes conflict with the adjustment of objective termwhich may cause oscillations near convergence point

This study investigates the TPO problem A mathematicmodel on the basis of the quadratic penalty function ofHNN is constructed Two strategies are presented to improvethe performance of TPO model one is the proportionalintegral derivative (PID) strategy that adaptively adjusts theparameter 120582 of the constrained term and parameter 120574 ofthe optimization objective term by controlling the valueof constraint violation 119892119896 the other is efficiency factorstrategy that an efficiency factor is introduced in model forprioritizing the constrained terms over the objective term

Data test shows that compared with the path of boomlength changing before optimization both the number of

Mathematical Problems in Engineering 3

Cylinder

position combination [0 0 0 0 0]state combination [1 1 1 1 1]

(a) Full retraction

zero Hole locations Pin

0 045 091


(b) A boom stretching state

fixed section telescopicsections


I II III IV V

(c) Full extension

Figure 1 Structure of a SPMB with five telescopic sections

sections that need to be moved and the total travels ofcylinder can be reduced by 10-30 after optimizationBoth the PID strategy and the efficiency factor strategyachieve good optimization effects It is found that the mainreason leading to the generations of low quality solution isthe oscillations being triggered during convergence processThe efficiency factor strategy is excellent at moderating theconflict between the constrained term and the objective termand restrains the oscillations successfully thereby The studyconsists of the following five parts

Section 1 IntroductionSection 2 SPMB mechanism and TPO problemSection 3 HNN method applied for TPO problemSection 4 Simulation analysis and discussionSection 5 Conclusion and prospect

2 SPMB Mechanism and TPO Problem

21 SPMB Mechanism Figure 1 illustrates an example ofSPMB with five telescopic sections which are denoted sepa-rately in signs of I II III IV and VThe length of each sectionis divided into four scales and each scale has a hole set forbeing locked by pin of the inner adjacent section Thereforethe different hole of the outer section being locked by thepin of the inner section determines the different stretchingof the inner section Let one-section length be defined as1 and the hole locations are 0 045 09 and 1 The fivetelescopic sections of the boom length may be expressed

as a combination 09 09 045 045 0 which means thatSections I and II are extended by 09 Sections III and IVare extended by 045 and Section V is not extended Tosimplify the expression of section length combination theholes locations 0 045 09 and 1 are replaced by four statesof ldquo1rdquo ldquo2rdquo ldquo3rdquo and ldquo4rdquo respectively Then the mentionedsection length combination is expressed in an array [3 3 2 21]

The array [1 1 1 1 1] in Figure 1(a) denotes the boomrsquos fullretraction state the array [3 3 2 2 1] in Figure 1(b) denotes theboomrsquos stretched state and the array [4 4 4 4 4] in Figure 1(c)denotes the boom in full extension

22 TPO Problem A telescoping step means the wholeprocedure being executed when a section is moved Theprocedure involves processes of the cylinder freely movingtowards aimed section and the cylinder driving the aimedsection to target position The procedure of the cylindermovement is similar to a round trip for example whenstretching a section the leaving trip is the cylinder drivingthe section forwards to target position (119878boom) whereas thereturning trip is the cylinder retracting itself backwards to theposition of next section (119878cylinder) When retracting a sectionthe leaving trip is the cylinder driving the section backwardsto target position (119878boom) whereas the returning trip is thecylinder stretching itself to the position of next section(119878cylinder) Therefore the optimization objective of TPO 119878totalincludes two parts the shortest cylinder telescoping path119878cylinder and the shortest boom section telescoping path 119878boom


Table 1 Effect comparison of the optimized paths with the non-optimized path

StepNO

Non-optimized

Drivensection

Optimization1

Drivensection

Optimization2

Drivensection

Optimization3

Drivensection

0 2 2 2 2 2 -- 2 2 2 2 2 -- 2 2 2 2 2 -- 2 2 2 2 2 --1 1 2 2 2 2 I 1 2 2 2 2 I 1 2 2 2 2 I 2 1 2 2 2 II2 1 1 2 2 2 II 1 1 2 2 2 II 1 1 2 2 2 II 2 1 1 2 2 III3 1 1 1 2 2 III 1 1 1 2 2 III 1 1 2 1 2 IV 2 1 1 1 2 IV4 1 1 1 1 2 IV 1 1 1 1 2 IV 2 1 2 1 2 I 2 1 2 1 2 III5 1 1 1 1 1 V 1 1 2 1 2 III6 1 1 1 1 2 V 2 1 2 1 2 I7 1 1 2 1 2 III8 2 1 2 1 2 IPath boom 36 27 18 18Path cylinder 36 36 27 27Total path 72 63 45 45Note Sign ldquoitalicrdquo denotes RBC

A boom with n-telescopic sections must run (2n-1) timesthe above procedures (telescoping steps) n times are forretracting n-sections full back and (n-1) times are for stretch-ing n-sections to target positions (because the endmostsection can save one step for direct stretching from initialstate to target state) if without optimization Then a boomwith 5-telescopic sections must run nine telescoping steps ifwithout optimizationThe repeated operations of proceduresmean heavy work and considerable time consumption IfTPO is performed the telescoping steps can be reduced byone or two Thus energy consumption and labor intensitycan be reduced and work efficiency can be promoted inengineering applications Table 1 shows an effect comparisonbefore and after TPO

In Table 1 the initial state combination is [2 2 2 2 2]and the target state combination is [2 1 2 1 2] Without anyoptimization RBC = [1 1 1 1 1] Eight steps are required tochange from initial boom length to target boom length Theboom path length (119878boom) is 36 and the cylinder path length(Scylinder) is 36 Thus the total path length (Stotal) is 72

With optimization 1 RBC = [1 1 1 1 2] six steps arenecessary 119878boom is 27 and the Scylinder is 36 Therefore Stotalis 63

With optimization 2 RBC = [1 1 2 1 2] and withoptimization 3 RBC = [2 1 1 1 2] Both RBCs only require foursteps 119878boom is 18 and Scylinder is 27 Thus Stotal is 45

Note that optimizations 2 and 3 are the optimal solutionsthe steps required are the least and the total paths are theshortest

23 RBC RBC is an extreme combination of retractionpositions for sections when changing the boom length Afterall sections are retracted to RBC state sections should extendsubsequentlyThe full retraction state [1 1 1 1 1] is always a validRBC for sections though the state is not an optimal RBCClearly the larger the RBC the shorter the path is becauseonly few retractions of sections being operated Therefore

the optimization goal is to find the maximum RBC so as toachieve the minimum distances of boom retracting from theinitial position to theRBCposition and boom stretching fromthe RBC position to the target position When the RBC isderived the telescoping path can be listed out subsequentlyas Table 1 listing

24 Single-Cylinder Travel Constraints during the TelescopingProcess (Constraints) Cylinder travel length must satisfyeach step in each section during the telescoping processRetraction or extension of any section is also performedWhether the length of its former section is within the travelof the cylinder must be considered The build-up evaluationequation of the cylinder travel allowance for each telescopingstep is as follows119892119896 = (C119895 [1] + 119862119895 [2] + + 119872119895 [119896] minus 1) le 0 (1)

In (1) j is the current telescopic step k is the cur-rent driven section 119862119895[1] 119862119895[119896-1] is the sum of theextension length of former (k-1) sections and 119872119895[119896]= max(119862119879119895[119896] and 119862119860119895[119896]) is the maximum extensionrequired in Section 119896 Herein 119862119879119895[k] is the target sectionlength and 119862119860119895[k] is the initial section length

25 Telescoping Path Definition (Optimization Objective)Figure 2 demonstrates the telescoping process of a 5-sectionboom and the total paths cylinder going through Blue cyclesindicate the initial and target positions of the sectionsOrangecycles indicate the RBC positions of the sections The whitering means the starting point of a cylinder (defined in zero)Values in brackets represent the potential energy height ofthe cylinder (absolute length) This measure cannot be over1 which is the cylinder travel limit at any time Solid linesrepresent cylinder path 119878cylinder and dotted lines show boompath 119878boom Their equations are provided below119878119905119900119905119886119897 = 119878119887119900119900119898 + 119878119888119910119897119894119899119889119890119903 (2)


(0)

Startpoint

-045

-045

0

-045

0

000 0

+045

(045) (045) (045) (09) (045)

(045)(045)(045)(0)(045)

1

2

1

2

2

2A1 A2 A3

V1 V2 V3 1

2A4

V4 2

2A5

V5

2T51T42T31T22T1

(0) (0) (045) (045) (09)

-045

-0450 0 -0

+045

+045

+045

+045

+0

Figure 2 Paths of SPMB telescoping process

119878119887119900119900119898 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111989511988910038161003816100381610038161003816) (3)

119878119888119910119897119894119899119889119890119903 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111990011989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111990011989511988910038161003816100381610038161003816) (4)

119860119895 is the initial section state 119879119895 is the target section state 119881119895is the RBC state and d is the hole location combination 119881 =[1198811 1198812 1198813 1198814 1198815] 119881o = [1198811 1198812 1198813 1198814 0] Figure 2 describesthat the boom sections retract from119860 = [2 2 2 2 2] to119881 = [1 12 1 2] and then stretch from the RBC to T=[2 1 2 1 2] Duringthe telescoping process the S119887119900119900119898 = 18 the S119888119910119897119894119899119889119890119903 = 27 andthe total path is 45

3 HNN Method Applied for TPO Problem

31 HNN Method In 1982 American physicist John JosephHopfield proposed a neural networkmodel which vigorouslypromoted the study of neural networks [5] Then Hopfieldand Tank (1985) successfully used this model to find thesolutions of the traveling salesman problem (TSP) [6] How-ever matching parameters is always difficult and improperparameters set leads to bad performance of HNN In orderto properly determine the weight coefficients of energy func-tion scientists have made a series of research Shirazi B andH S (1989) used a matrix method to analyze the dynamics ofcontinuous HNN and thus analyze the basic characteristicsadvantages and disadvantages of the model [7] Aiyer etal (1990) explained why the Hopfield network often fallsinto an invalid solution in the TSP problem by means ofanalyzing matrix eigenvalues and hypercube mapping Thenthey modified the energy function and derived parameterssetting principles to ensure that the network converges toa valid solution [8] Abe S (1993) analyzed the hypercubevertex condition as it becomes a local minimum The author

also provided a method for suppressing the inferior solutionon the basis of which weights and coefficients of the energyequation are set [9] Sun S et al (1995) simplified theequation of Aiyer et alrsquos equation and demonstrated thevalidity of the penalty parameter determination based onAiyerrsquos matrix eigenvalue analysis method Their equationalmost achieved a 100 valid solution obtained from the10-city TSP problem [10] Subsequent researchers such asZhang J et al (1996) tested Sunrsquos network but it was unableto obtain 100 valid solutions an only calculated 70 validsolutions after extensive examination of a five-city TSP [11]However Sunrsquos equation is still a simple and efficient formfor TSP problems Pedro M Talavan and Javier Yanez (2002)introduced a parameter setting method for a stable conditionanalysis using the valid solution of an energy equation Oncethe parameters are determined by the analytical methodany stable point becomes a valid path for TSP [12] Effatiand Baymain (2005) proposed the parameters chosen fromthe constrained differential equations [13] Effati S et al(2007) demonstrated that the neural network in the formof quadratic penalty function is a Lyapunov function withuniform convergence [14]

Equation of Sun Shouyu for TSP is listed as follows [10]

119864 = 1198602 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1198612 119898sum

119894=1

( 119899sum119909=1

119881119909119894 minus 1)2+ 1198632 119899sum119909=1 119899sum119910=1 119898sum119894=1119881119909119894119889119909119910119881119909119894119881119910119894+1

(5)

A and 119861 are parameters of the constrained terms D isparameter of the objective term Determinations of A Band119863 follow principles of Aiyerrsquos matrix eigenvalue analysisSeeing that the principles of Aiyer are complicated we applythe quadratic penalty function to construct model but takeadaptive method to determine parameters


32 Permutation Matrix Representation of Section States Apermutation matrix is used to represent the sections andthe corresponding positions of pins where 119881119909119894 is the pinnedposition 119894 of the Section119909 of RBCV 119909 = 1 119899 with 119899 as thenumber of sections 119868 = 1 119898 with119898 as the number of pinholes A is the initial state matrix T is the target state matrixV is the RBC state matrix and 119889 is the pinhole location arraysuch as d = [0 045 09 1]1015840

119881 =(((

1 0 0 01 0 0 00 1 0 01 0 0 00 1 0 0)))

119881 times 119889 =(((

000450045)))

(6)

The given example matrix 119881 denotes that RBC = [1 1 2 1 2]V times d denotes the extension length of boom sections

33 Energy Equation of TPO The energy function is givenbelow

119864TPO = 1198612 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1205822

sdot 119899sum119896=1

[S119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)]2 + 1198881205742sdot 119899sum119909=1

[[(119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894)2

+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119894119889119894)2+ ( 119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)2+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119900119894119889119894)2]]119872119896 = max( 119898sum

119894=1

119860119896119894119889119894 119898sum119894=1

119879119896119894119889119894) k = 1 119899g119896 (119881) = [ 119896sum

119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1]

119878119896 = 0 g119896 le 01 g119896 gt 0

(7)

n is sections number and m is holes number of each sectionsupposing the hole distributions on each section are the sameThere are total three terms in (7)

A Term B Equality constrained term is the row-constraintof permutation matrix 119881 The term defines that each row inmatrix119881 has one and only one element ldquo1rdquo and the remainingelements are ldquo0rdquo Its physical meaning is that each section hasone and only one hole being pinned

B Term 120582 Inequality constrained term is the row-constrained term of RBC (V times d) which denotes theoutstretched length of the section This term is composed ofnumber 119899 inequality constraints that are associated with oneanother Its physical meaning is defined as the constraintson each section and on the stretching of each step Duringthe retraction and extension of Section x the sum of theextension length of all previous (x-1) sections plus themaximum extension required length of current Section 119909must be less than or equal to ldquo1rdquo (total travel of a singlecylinder) From the second term in (7) we can see that

k = 1 when driving Section I its maximum telescopiclength should be less than the cylinder travel lengthldquo1rdquo 1198921 = (1198721 minus 1) le 01198781 equiv 0 (8)

k = 2 when driving Section II the current extensionof Section I plus the maximum telescopic length ofSection II should be less than the cylinder travellength ldquo1rdquo

1198922 = ( 2sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +1198722 minus 1) le 0 (9)

k = n when driving Section N the current extensionsof Sections I to (N-1) plus the maximum telescopiclength of Section N should be less than the cylindertravel length ldquo1rdquo

119892119899 = ( 119899sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119899 minus 1) le 0 (10)

C Term 120574 Optimization objective term is defined as the sumof squares of paths lengths (S119888119910119897119894119899119889119890119903 and S119887119900119900119898) of each sectionretracting from the initial position to the RBC position andextending from the RBC position to the target position Pathscan only be minimized when the RBC of matrix 119881 takes amaximum

B 120582 120574 are term parameters of (7) where 119888 is the efficiencyfactor When c = 1 efficiency factor does not work Equation


(11) reveals the relationship between the input 119880119894 and theoutput 119881119894 of neurons in which the activation function is thehyperbolic tangent function Furthermore 120572 is the ramp

119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)

34 Dynamic Differential Equation Dynamic differentialequation is derived as follows

Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)

Let 119892119896 = (sum119896119909=1sum119898119894=1 119881(119909minus1)119894119889119894 + 119872119896 minus 1) the second term of(13) can be expanded as followsΔ119880120582 = (120582111987811198921 + 120582211987821198922 + + 120582119899119878119899119892119899) 1198891198941015840= [(12058221198782 + 12058231198783 + + 120582119899119878119899) (1198811 times 119889)+ (12058231198783 + 12058241198784 + + 120582119899119878119899) (1198812 times 119889) + + 120582119899119878119899 (119881119899minus1 times 119889) + 12058211198781 (1198721 minus 1)+ 12058221198782 (1198722 minus 1) + + 120582119899119878119899 (119872119899 minus 1)] 1198891198941015840

(14)

Equation (14) indicates that when being at the RBC statethe restraint for the extension of Section I (1198811times d) is at itsstrongest which is (1205822S2 + 1205823S3 + + 120582119899S119899) The restraintfor the extension of Section II is slightly weak at (1205823S3+ 1205824S4 + + 120582119899S119899) No restraint exists for the extensionof Section N The given explanation is consistent with thereality that the endmost section (Section N) can arbitrarilyretract and stretch since it will not influence themovement ofother sections Therefore Section N is the most free On thecontrary the length changing of the foremost section (SectionI) will have a subsequent effect on the movements of all othersections Thus its freedom is the smallest and its restraint isthe strongest Telescoping performed in sequence simply isto eliminate the associated influence on each section In (14)Section I is subjected to the biggest penalty to minimize thestretching length (1198811times d) and release cylinder travel space forthe retracting and stretching of subsequent sections

Equation (15) is the dynamic updating equation where 119897119903is iteration step size119880 (t + 1) = 119880 (119905) minus 119897119903 times Δ119880 (15)

35 PID Adaptive Parameter Adjustment Strategy Matchingpenalty parameters is always difficult The consequence ofimproper selection of parameters is that the convergence istrapped in localminimumand could not get optimal solution

In order to escape the local optimal solution there aregenerally some categories of strategies one is to adjust theparameter and to change the weight values of the neuralnetwork Since the weight values determine the shape ofenergy surface the gradient descent path is changed therebyPID adaptive parameter adjustment is in this way whichadaptively adjusts the descent path towards the lower pointof energy loss by changing the shape of energy surface timely

Other strategies are like that when the parameters and theinput are constant the shape of energy surface is determinedIf the energy loss stops at a saddle point there might havebeen a force to push it continuously descending and themomentum strategy is in this way If the energy loss is hardto converge to a balance point stably the iteration step sizecan be regulated to help convergence and the learning rateadjustment strategy is in this way [15]

Some other strategies are proposed by researchers topromote the searching effect of network For example thehill jumping algorithm divides energy function 119864 into twopartsmdash1198641 and 1198642 Energy functions119864 1198641 and 1198642 are alterna-tively run and their weights and bias are recordedTheweightand bias of one energy function are set as the new startingpoint of another energy function Finally a global minimumpoint is promisingly reached [16] In addition there arenoise gradient strategy [17] Stable Manifold Theorem [18]noise chaotic neural network [19] noise vector strategy [20]method of increase training times [21] and so on

Principles of PID adaptive parameter adjustment are asfollows

Associate Δ120582 with 119892119896Δ120582119896 (119905) = 119870119901120582 sdot 119892119896 (119905) + 119870119894120582 sdot sum119905

119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)

120574 (119905 + 1) = 120574 (119905) minus 120583120574 sdot Δ120574 (119905) sdot 120574 (119905) (19)119870119901 is the proportional coefficient 119870119894 is the integral coeffi-cient and119870119889 is the differential coefficient 120582 is the parameterof the constrained term of (7) equations (16) and (17) arethe setting laws of 120582 and the PID adaptive adjustment of120582 is based on the control of constraint violation 119892 and isgradient-rising 120574 is the parameter of the objective term of (7)equations (18) and (19) are the setting laws of 120574 and the PIDadaptive adjustment of 120574 is based on the control of constraintviolation 119892 and is gradient-falling Figure 3 illustrates that1205821 1205825 are the PID adaptive curves for the constrained term


100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455

Figure 3 Adaptive adjustment curves for 120582 and 120574and 1205741 1205745 are the PID adaptive curves for the objectiveterm

The adaptive parameter setting can often obtain goodoptimization effect However the PID parameters controlmay have conflict with the trend of gradient descendingwhen the two kinds of adjustments conflict on some equi-librium point oscillations are triggered Thus the networkcannot converge to balance point stably Moreover theadaptive parameter setting is boundedThuswhenparameteradjustment reaches the upper or the lower limit adjustmentstrength does not change any more thereby impeding thenetwork converging to the global optimum Basing on abovedifficulties the further strategy for improvement is proposed

36 Efficiency Factor Strategy As a problem of the opti-mal solutions distributing on boundary of constraints theadjustments of constrained terms sometimes conflict with theadjustment of objective terms which may cause oscillationson equilibrium point In addition as the analysis in 35the PID parameter adjustments may violate the gradientdescent trend of the network which also leads to oscillationson balance point The main reason triggering oscillationsis that the constraint parameter controls and the objec-tive parameter controls are hard to balance For examplewhen the constrained term dominates the descent energyloss descends toward constraints satisfaction whereas theoptimization objective might not be satisfied Therefore inthe next iteration regulating strength for the objective termgrows larger thus the objective term dominates the descentand drives it toward the optimum whereas the convergencemight break away the domain of feasible solution and theconstraints might not be satisfied Such repeated processeslead to oscillations on the equilibrium point

To solve the problem one strategy is to convert theconstrained optimization to multiobjective optimization

minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ

Figure 4 Curve of efficiency factor 119888119896However the higher the dimension of the multiobjectiveoptimization the more difficult the optimization model is tobe calculated [22] Therefore this study takes another waythat introduces an efficiency factor into the energy functionto prioritize the constrained terms over the objective term

The second term of (7) is the term for associated con-straints 119888119896 is the efficiency factor of Section k k = 1 119899119892119896 is the constraint violation of Section k which belongs to[minus1 0] If Section 119896 is fully retracted then the cylinder hasfull travel allowance and ldquo1rdquo for driving Section k 119892119896 will bendash1 If Section 119896 is fully extended then the cylinder has notravel allowance ldquo0rdquo for driving Section k and 119892119896 will be 0Generally 119888119896 should be normalized but because 119892119896 has beenlimited in the range of [minus1 0] and 119888119896 has been limited to [0 1]thereby 119888119896 does not need to be normalized here119888119896 = 0 119892119896 gt 0119888119896 = 1 + 119892119896 119892119896 le 0 (20)

119888 = 119899prod119896=1

119888119896 (21)

Equation (20) indicates that when 119892119896 gt 0 convergence isbeyond the feasible domain and then 119888119896 = 0 When ndash1 le 119892119896 le0 convergence is within the feasible domain and then 119888119896 isvariable with 119892119896 changing When 119892119896 = 0 network convergesto the boundary of the feasible domain which is the optimalsolution and then 119888119896 = 1 Figure 4 shows the curve of 119888119896 c isthe multiplication of 119888119896 as presented in (21)

4 Simulation Analysis and Discussion

41 Effect of TPO Let n=5m=4 in (7) that is a SPMBhas fivetelescopic sections and four holes arranged on each sectionThe holes locations are the same for all sections Parametersare set as shown in Table 2 Input the initial state A=[2 2 2 22] and the target state T=[2 1 2 1 2] and run the TPO HNNprogram to search for RBC state V

Figure 5 shows the telescoping processes with four RBCstates In these figures the color blocks represent the sectionsand the blue dotted line is the single-cylinder travel ldquo1rdquoBecause the travel of cylinder driving section (119878boom) can


Table 2 Parameter setting of simulations

HNN 120582 of PID 120574 of PID Momentum Adagrad119861 = 5 119861 = 5 119861 = 5 119861 = 5 119861 = 5120583 = 00001 120583120582 = 00001 120583120574 = 00001 119897119903 = 0001 119897119903 = 01120582 = [5 5 5 5 5] 119870119901120582 = 10 119870119901120574 = 01 120582 = [5 5 5 5 5] 120582 = [5 5 5 5 5]120574 = [5 5 5 5 5] 119870119894120582 = 01 119870119894120574 = 001 120574 = [5 5 5 5 5] 120574 = [5 5 5 5 5]119870119889120582 = 1 119870119889120574 = 1 120583119898 = 09120582(0) = 0 120574(0) = 1000

1 2 3 4 5 6 7 8 9 10initminusminusRBCminusminustarget

Telescopic Steps (Shortest Path Length36 )

Section1Section2Section3

Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5


Telescopic Steps (Shortest Path Length18)

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]

Figure 5 Scheduled telescoping paths from initial state A=[2 2 2 2 2] to target state T=[2 1 2 1 2]


Table 3 Test results for TPO-HNNmodel

data HNN HNN+c HNN+PID HNN+PID+c HNN+Momentum HNN+Adagrad

Run 25 times Optimalvalid time Optimal

valid time Optimalvalid time Optimal


valid time

A = [2 2 2 2 2] 719 0511 1325 0499 1222 1043 1021 1119 520 0535 322 0554T = [2 1 2 1 2]A = [3 2 3 2 1] 17 0531 320 0528 115 1081 514 1082 26 0590 37 0510T = [1 2 2 2 2]A = [2 1 3 3 1] 25 0553 620 0549 513 1064 519 1164 44 0537 15 0494T = [1 2 2 2 2]A = [1 2 4 2 1] 210 0565 625 0551 517 1120 720 1103 36 0526 511 0517T = [2 2 2 2 2]A = [2 1 2 1 2] 115 0530 824 0516 515 1024 1014 1145 417 0533 315 0526T = [2 3 2 2 1]Averagepercentage

10 29 22 30 14 1245 95 65 70 42 48

identify path length well we use 119878boom instead of the totaltravel of cylinder (119878total) to indicate the optimization effect

If the RBC states are not well optimized or not optimizedthe telescoping paths will require six steps and 119878boom will be36 as shown in Figures 5(a) and 5(b) If the RBC states arewell optimized they will just require four steps and 119878boomwill only be 18 as shown in Figures 5(c) and 5(d) Theefficiency of boom length changing increases obviously afterwell optimization

42 Improvement of the TPO HNN Model The probabilitythat energy loss converges to the saddle point is large becausemost natural objective functions have exponential saddlepoints Such points are unstable implying invalid solutionsMomentum strategy helps accelerate gradient descent incertain directions and suppress oscillation [23]

V (119905) = 120583119898 times V (119905 minus 1) + 119897119903 times 119889119906 (119905 minus 1) (22)119906 (119905) = 119906 (119905 minus 1) minus V (119905) (23)

The given formula is the iterative formula of momentumstrategy where v(tndash1) is the direction of the last iteration 120583119898is the momentum coefficient 119897119903 is the learning rate (iterationstep size) du(t) is the gradient and u(t) is the output [24]

The descent can easily stuck in the saddle point therebyleading to divergent oscillation with a fixed learning rateThe adaptive gradient algorithm (Adagrad) adaptively adjuststhe learning rate 120576 is a very small constant that prevents thedenominator from becoming zero [25]119886119888119888 (119905) = 119886119888119888 (119905 minus 1) + (119889119906 (119905 minus 1))2 (24)

119906 (119905) = 119906 (119905 minus 1) minus 119897119903radic119886119888119888 (119905) + 120576119889119906 (119905 minus 1) (25)

In order to improve the performance of TPO modelthese two strategies are used in TPO model to compare theeffects with PID strategy and efficiency factor strategy In

the following six strategies are simulated respectively HNNHNN with efficiency factor HNN with PID adjustmentHNN with PID and efficiency factor mixed HNN withmomentum and HNN with Adagrad Five groups of dataare tested and each group runs 25 times The iteration is2000 and 1198810 is randomly generated Parameter settings areas Table 2 and the results are listed in Table 3

When parameters are set as constants the performance ofHNNmodel is not good During the 25 runs of program theobtained valid solutions percentage is lower than 50 How-ever after introducing efficiency factor in the HNN modelthe performance is evidently improved and the obtainedaverage valid solution percentage increases to 95 The PIDadaptive parameter adjustment strategy also performs welland the obtained average valid solution percentage reaches65 The HNN with both PID and efficiency factor strategyalso improves the results well and the obtained average validsolution percentage is 70

Furthermore the average optimal solution percentagesobtained by the efficiency factor strategy by the PID strategyand by the PID and efficiency factor mixed strategy are thebest Thus either the PID strategy or the efficiency factorstrategy is effective in promoting the solution qualities

As a problem that its high-quality solutions are scatteredon the boundary of a feasible domain as Keanersquos bumpproblem [26] the oscillations of TPOmodel are caused by themutual conflicts of the constrained terms and the objectiveterms Besides the PID adaptive parameters control mighthave contradiction with the trend of gradient descendingwhich brings the convergence oscillations as well whilethe efficiency factor strategy is workable to balance thecontradictions of the constrained terms and the objectiveterms

Although the strategy of momentum and the strategy ofchanging learning rate are efficient to help the convergenceescape saddle point they do not work well here sincestopping at saddle point is not the reason of triggering theoscillations while TPO model converging


Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5

10Sum g(sum of gapsminus275)

(a) 119892k control of HNN+PID strategy

0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5

(b) 119892k control of HNN+PID+c strategy

Figure 6 The constraint violation (119892) changes during convergence processIn addition PID adaptive parameter adjustment strategy

has a slightly higher complexity than othermethodsThus itscalculation time is slightly longer than other methods

43 Analysis of TPOCalculation Process Theefficiency factorstrategy plays an important role in promoting the qualitiesof solutions derived from Table 3 Through comparing theconvergences before and after efficiency factor strategy beingadded in the model we can know how the efficiency factorstrategy works

The parameter settings are as in Table 2 the iterationis 5000 the initial RBC (1198810) is fixed and the same initialconditions are applied for the compared strategies Test dataare A = [2 1 3 3 1] T = [1 2 2 2 2] HNN with PIDadaptive parameters HNN with PID adaptive parametersand efficiency factor mixed HNN with constant parametersand HNN with efficiency factormdashthese four strategiesmdasharetested

HNNPID strategy adaptively adjusts constrained param-eter 120582 and objective parameter 120574 by controlling the constraintviolation 119892 to close the zero which is the boundary of thefeasible domain 1198921 1198925 are the constraint violations of thefive sections and 119878119892 is the sumof1198921 1198925 Figure 6(a) depictsthe 1198921 1198925 and 119878119892 changes in the process of the HNN PIDnetwork converging with 1198810 as the initial input When theiteration is over 119878119892 obtains high cylinder travel allowancendash275 which indicates that the path is not fully optimizedFigure 6(b) demonstrates the 1198921 1198925 and Sg changes in theprocess of the HNN PID and efficiency factor mixed networkconverging with the same 1198810 as the initial input When the

iteration is over 119878119892 obtains little cylinder travel allowancendash05 which indicates that the path is well optimized when theefficiency factor is being added in the HNN PID model

Figures 7(a) and 7(b) compare the solution V rsquos conver-gence process under the given conditions Figure 7(a) showsthat 1198814 has oscillations triggered between ldquo1rdquo and ldquo2rdquo afterthe program iterates approximately 2500 times The reasonis that when iterating to a certain extent value of 11988141 andvalue of 11988142 are almost equal and the definition of matrix 119881is based on a ldquomaximum pickrdquo principle where themaximumvalue among the row elements is picked out as the output andall else values are ignored Thus when two similar elementsexist in the same row a selection jumps between these twoelements Figure 7(b) shows that when efficiency factor isintroduced into the PID strategy the oscillations of1198814 do nothappen and it converges stably on ldquo2rdquo The efficiency factorprioritizes the constrained terms over the objective terms andall the termsdonot have to competewith each otherThus theoscillations are suppressed and the generations of high qualitysolutions are increased

Figure 8(a) describes the scheduled telescoping path insolution V = [1 1 1 1 2] which is derived by PID strategy Theprocess requires eight steps from the initial state to the RBCstate and then to the target state S119887119900119900119898 is 405 Figure 8(b)illustrates the scheduled telescoping path in solution V = [11 1 2 2] which is derived by PID and efficiency factor mixedstrategyThe process requires seven steps from the initial stateto the RBC state and then to the target state S119887119900119900119898 is 315Theefficiency factor strategy is useful to help the convergence tooptimal solution


05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608

(a) V convergence of HNN+PID

05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608

(b) V convergence of HNN+PID+c

Figure 7 The RBC 119881 converging processes


Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25

(a) With HNN+PID strategy


Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25

(b) With HNN+PID+c strategy

Figure 8 Scheduled telescoping path from initial state A = [2 1 3 3 1] to target state T = [1 2 2 2 2]Figure 9 demonstrates the energy loss curves with PID

strategy before and after introducing in efficiency factorunder the same initial conditions Blue curve is the energyloss of PID strategy which converges to minimum 10482corresponding to a valid solution V = [1 1 1 1 2] Redcurve is the energy loss of PID mixed with efficiency factorstrategy which converges tominimum 59212 correspondingto an optimal V = [1 1 1 2 2] The comparison indicatesthat efficiency factor is helpful in converging to the optimalpoint

Figure 10 demonstrates the energy loss curves of HNNwith constant parameters strategy before and after introduc-ing efficiency factor under the same initial conditions Bluecurve is the energy loss which converges to minimum 65127

corresponding to an invalid solution V = [1 2 2 1 2] Redcurve is the energy loss with efficiency factor strategy whichconverges to minimum 030891 corresponding to an optimalV = [1 1 1 2 2] The comparison indicates that the efficiencyfactor greatly enhances the descent searching strength thusmaking the energy loss converging to a global minimumWhile having no efficiency factor strategy the energy lossshould have oscillated around an invalid solution

In summary both PID strategy and efficiency factorstrategy are effective in improving the performance of theTPO model The efficiency factor strategy has its advantageson balancing the conflicts between the constrained terms andobjective terms so it can lead to stable convergence efficientlyand can get high quality solutions thereby


Energy Function Curve (Optimum Energy10482 59212)

PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration

Figure 9 Energy curves of HNN+PID and HNN+PID+c

0 500 1000 1500 20000

20

40

60

80

100

120

140

160

180Energy Function Curve (Optimum Energy65127 030891)

Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]

Figure 10 Energy curves of HNN and HNN+c

5 Conclusion and Prospect

This study investigates the practical engineering optimizationproblem TPO of SPMB TPO aims to obtain the maximumRBC During the SPMB telescoping process the movementof each section is associated with each other and mutu-ally restrained that is multiple associated constraints areinvolved in TPO problem TPO is a strongly constrainedproblem with the optimal solutions scattered on constraintboundaries In a word TPO aims to obtain the largest RBC

thus the scheduled telescoping steps can be the least and thecylinder telescoping paths can be the shortest when SPMBchanges its boom length

The energy function model of HNN can mitigate thecomplexity of constraints processing thus this article con-structs the TPO mathematical model in quadric penaltyfunction form of HNN In the energy function seeing thatdetermining the penalty parameters is difficult a PID strategyis proposed that adaptively adjusts the penalty parameters120582 of the constrained term and the penalty parameters 120574of the optimization objective term by controlling constraintviolation value 119892119896 Moreover the 120582 is set as gradient-risingand the 120574 is set as gradient-falling according to the trend ofthe dynamic equation gradient descending The overlappingregion of 120582 and 120574 is the place for the optimization searchingof the network where the larger the optimization searchingarea the more likely converging to high-quality solution

TPO belongs to the optimization category that optimalsolutions scattered on boundary of feasible domain Theconstrained terms and the objective terms are sometimesmutually exclusive so they are difficult to be balancedThe oscillations being triggered sometimes on equilibriumpoint make the network hardly converge to global optimalBeside although the PID strategy is effective in drivingthe convergence towards the valid or the optimal solutionsthe PID adaptive parameters control might have contradic-tion with the trend of gradient descending which bringsthe oscillations on equilibrium point as well Oscillationis the reason that the network could not converge effi-ciently

The introduction of efficiency factor that prioritizes theconstrained terms over the objective terms is efficient to solvethe oscillation problem and gets the best generations of bothvalid and optimal solutions

The simulation result shows that compared with theboom length changing before optimization both the numberof sections that need to be moved (scheduled telescopingsteps) and the total travels of the cylinder can be reduced by10-30 after optimization

Compared to the effect of the HNN with constantparameters strategy the HNN PID strategy can improve thevalid solutions percentage from 50 to 65 while the HNNefficiency factor strategy is more efficient and improves thevalid solutions percentage from 50 to 95 The optimalsolutions percentage obtained by efficiency factor strategy isthe highest as well In summary the efficiency factor strategyis excellent at balancing the conflicts between the constrainedterms and objective terms

The PID strategy and the efficiency factor strategy areuniversal that can be applied for other problems solving TheHNNmethod for TPO problem is a universal model that canbe applied to calculate the TPOs under various conditionsfor example the boom sections n the holes number 119898 andthe holes location combination 119889 on each section are variablenot limited by n=5 andm=4 d= [0 045 09 1] of the examplebeing illustrated in the article In addition the HNNmethodcan be used to the TPOs of complex SPMB mechanisms forexample the holes number and the holes locations are variousfor every section


Although the HNN model with PID strategy and theHNNmodel with efficiency factor strategy both obtain goodresults there are flaws that should be attended

(I) PID adaptive parameters adjustment strategy hasto determine the proportional coefficient 119870119901 theintegral coefficient 119870119894 and the differential coefficient119870119889 thus it is not the fully self-adaptive strategy

(II) The searching algorithm basing on gradient descentmethod is hard to obtain 100 valid solutions and thegeneration of optimal solutions is not high generally

In future the genetic algorithm (GA) could be used tocalculate TPOs seeing that GA only requires the optimizedobjectives being computable and no need being continuousand differentiable besides GA is based on population evolu-tion that converges to global optimums so it can overcomethe weaknesses of traditional algorithms based on gradientoptimization

Data Availability

The [DATA programs] data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] C Zhan Q Liu J Guo et al ldquoSingle-cylinder pin-type tele-scopic boom track optimized control method and controlsystem thereofrdquo Tech Rep WO2011038633 2011

[2] Y Mao and K Chen ldquoSequential telescopic path optimizationmethod of single-cylinder pin-type multi-section boomrdquo TechRep CN106744386B 2018 (Chinese)

[3] Y Mao and K Chen ldquoAn efficient sequential telescopic pathoptimization method of single-cylinder pin-type multi-sectionboomrdquo Tech Rep CN106744389B 2018 (Chinese)

[4] U-P Wen K-M Lan and H-S Shih ldquoA review of Hopfieldneural networks for solving mathematical programming prob-lemsrdquo European Journal of Operational Research vol 198 no 3pp 675ndash687 2009

[5] J J Hopfield ldquoNeural networks and physical systems withemergent collective computational abilitiesrdquo Proceedings of theNational Acadamy of Sciences of the United States of Americavol 79 no 8 pp 2554ndash2558 1982

[6] J J Hopfield and DW Tank ldquoNeural computation of decisionsin optimization problemsrdquo Biological Cybernetics vol 52 no 3pp 141ndash152 1985

[7] B Shirazi and S Yih ldquoCritical analysis of applying Hopfieldneural net model to optimization problemsrdquo in Proceedingsof the IEEE International Conference on Systems Man andCybernetics vol 1 pp 210ndash215 1989

[8] S Aiyer M Niranjan and F Fallside ldquoA theoretical inves-tigation into the performance of the Hopfield modelrdquo IEEETransactions on Neural Networks and Learning Systems vol 1no 2 pp 204ndash215 1990

[9] S Abe ldquoGlobal convergence and suppression of spurious statesof the Hopfield neural networksrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 40 no4 pp 246ndash257 1993

[10] S Shouyu and J Zheng ldquoA modified algorithm and theoreticalanalysis for hopfield network solving TSPrdquo Acta ElectronicaSinica 1995 (Chinese)

[11] J Zhang ldquoTheoretical analysis and improvement of the neuralnetwork to solve the TSP problemrdquo Journal of Xidian Universityvol 23 1996 (Chinese)

[12] P M Talavan and J Yanez ldquoParameter setting of the Hopfieldnetwork applied to TSPrdquoNeural Networks vol 15 no 3 pp 363ndash373 2002

[13] S Effati and M Baymani ldquoA new nonlinear neural networkfor solving convex nonlinear programming problemsrdquo AppliedMathematics and Computation vol 168 no 2 pp 1370ndash13792005

[14] S Effati and M Jafarzadeh ldquoA new nonlinear neural networkfor solving a class of constrained parametric optimizationproblemsrdquo Applied Mathematics and Computation vol 186 no1 pp 814ndash819 2007

[15] S Ruder ldquoAn overview of gradient descent optimization algo-rithmsrdquo 2016

[16] R Wang S Guo and K Okazaki ldquoA hill-jump algorithm ofHopfield neural network for shortest path problem in commu-nication networkrdquo Soft Computing vol 13 no 6 pp 551ndash5582009

[17] L Wang and F Tian ldquoNoisy chaotic neural networks forsolving combinatorial optimization problemsrdquo in Proceedings ofthe IEEE-INNS-ENNS International Joint Conference on NeuralNetworks (IJCNN rsquo00) vol 4 pp 37ndash40 IEEE Como Italy2000

[18] L Perko Differential Equations and Dynamical SystemsSpringer 2001

[19] R Ge F Huang C Jin and Y Yuan ldquoEscaping from saddlepoints online stochastic gradient for tensor decompositionrdquoJournal of Machine Learning Research vol 40 no 2015 2015

[20] A El-Bouri S Balakrishnan and N Popplewell ldquoA neuralnetwork to enhance local search in the permutation flowshoprdquoComputers amp Industrial Engineering vol 49 no 1 pp 182ndash1962005

[21] H Qu Z Yi and H Tang ldquoImproving local minima ofcolumnar competitive model for TSPsrdquo IEEE Transactions onCircuits and Systems I Regular Papers vol 53 no 6 pp 1353ndash1362 2006

[22] YWang Z Cai G Guo andY Zhou ldquoMultiobjective optimiza-tion and hybrid evolutionary algorithm to solve constrainedoptimization problemsrdquo IEEE Transactions on Systems Manand Cybernetics Part B Cybernetics vol 37 no 3 pp 560ndash5752007

[23] M Nielsen ldquoNeural networks and deep learningrdquo 2016 httpneuralnetworksanddeeplearningcom

[24] N Qian ldquoOn the momentum term in gradient descent learningalgorithmsrdquo Neural Networks vol 12 no 1 pp 145ndash151 1999

[25] J Duchi E Hazan and Y Singer ldquoAdaptive subgradient meth-ods for online learning and stochastic optimizationrdquo Journal ofMachine Learning Research vol 12 no 7 pp 257ndash269 2011

[26] M Schoenauer and Z Michalewicz ldquoBoundary operators forconstrained optimization problemsrdquo in Proceedings of the 7thInternational Conference on Genetic Algorithms T Baeck Edpp 322ndash329 Morgan Kaufmann Publishers San Mateo CalifUSA 1997

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the multiple sections must await to be retracted one by one insequence from initial position to backmost position and thenwait to be stretched in reverse sequence from the backmostposition to target position while changing boom length fromone state to another When telescoping one section a setof complex procedure (telescoping step) must be operatedmainly involving processes of cylinder freely moving towardsaimed section and cylinder driving the aimed section towardstarget position Provided that a boom has 119899 sections (2n-1) telescoping steps should be operated when changing theboom length where (n-1) steps are for retracting sectionsand 119899 steps are for stretching sections If optimization isperformed the telescoping steps can be reduced by one ortwo and then the work efficiency can be evidently improved

TPO of SPMB is a practical engineering problemAlthough SPMB is widely applied in machinery and equip-ment research on TPO is lacking

Reference [1] (2012) described the TPO problem firstin public and optimized a five-section SPMB in a nestedprogram The principle was that judging the sections fromthe outermost section to the innermost section if therequired cylinder travel exceeded the offered cylinder travelretracted the section fully otherwise it kept the sectionunmoved [1] Reference [1] is enumeration method andmakes judgment only between two states of ldquoretract fullyrdquoand ldquokeep unmovedrdquo Thus the effect of optimization is verylimited

Mao Y et al (2018) proposed a simplified Permutationand Combination (PampC) method that chose three states foreach section to participate PampC those were ldquohold at initialpositionrdquo ldquomove to target positionrdquo and ldquoretract to full backrdquo(actually there are usually more states than these three Ifthere are number119898 holes on a section there will be number119898 stretched states for the section) Took these three states ofall sections to form state combinations and to participate PampCtogether then eliminate the state combinations exceedingcylinder travel allowance (invalid solutions) by function eval-uations and finally pick out the state combinations havingthe shortest paths (optimal solutions) from the remainingstate combinations (valid solutions) [2] PampC algorithmbelongs to enumeration category which can get optimalsolutions in short time when boom sections are few But ifthe number of boom sections increases its calculation willgrow up exponentially Besides the simplified handling thatonly three states for each section are chosen to form statecombinations of all sections cannot describe all telescopingpath possibilities

Reference [3] (2018) is another method to calculatethe TPO The method evaluated permitted cylinder travelallowance starting from the endmost section back to theforemost section Through two function evaluations of max-imum permitted travel allowance and minimum permittedtravel allowance it was decided whether opening the sidefunction evaluations or retracting one section full back andthen continuing downward evaluations [3]

References [1ndash3] are enumeration methods in nature andare suitable for small-scale problems because their logic willbe increasingly complicated with problem scales enlargingBesides thesemethods only pick out limited number of states

of the holes locations to make judgments in order to simplifycalculations and cannot reflect all the path possibilities sotheir optimization effects are not very well

TPO refers to the fact that when telescoping a multisec-tion boom from an initial state to a target state the numberof telescoping steps is the smallest and the total travelsof the driving cylinder is the shortest TPO is essentiallythe shortest-path scheduling problem of combination opti-mization problem (COP) TPO aims to work out maximumretraction backmost combination (RBC) that is the RBCposition always exists while all sections retract from theinitial position combination After retracting sections to theRBC position sections should be stretched to target positioncombination subsequently RBC length is limited in thefull travel of the long cylinder that is set as ldquo1rdquo The TPOproblem has the following features Movements of sectionsshould be operated in sequence When one section is readyto be retracted or stretched the current sufficiency of thetravel allowance of the driving cylinder should be consideredMoreover one sectionrsquos movement can affect other sectionsrsquomovability Thus TPO contains multiple constraints thatassociate with each other

Small-scale COPs can be solved by exact algorithms suchas dynamic programming branch definition and enumer-ation Heuristics can quickly calculate approximate optimalsolutions for large-scale complex problems but their opti-mality cannot be guaranteed Heuristics include neighboralgorithms simulated annealing evolutionary algorithmsand neural networks

The Hopfield neural network (HNN) has advantages insolving COPs In practical applications when converting theobjective function of the optimization problem to the energyequation of HNN to map variables to neural states in thenetwork HNN can be used to solve COPs That is when theneural state of the network tends to equilibrium the energyequation of the network converges to minimum the net-workrsquos convergence from initial state to steady state illustratesthe optimization calculating process of the objective function[4]

The energy equation is convenient to constraints dis-posing yet HNN has following difficulties (I) the penaltyparameters of energy equation are difficult to determine(II) the convergence is often trapped in local minimum(III) for problems of the optimal solutions distributing onboundary of constraints the adjustments of constrained termsometimes conflict with the adjustment of objective termwhich may cause oscillations near convergence point

This study investigates the TPO problem A mathematicmodel on the basis of the quadratic penalty function ofHNN is constructed Two strategies are presented to improvethe performance of TPO model one is the proportionalintegral derivative (PID) strategy that adaptively adjusts theparameter 120582 of the constrained term and parameter 120574 ofthe optimization objective term by controlling the valueof constraint violation 119892119896 the other is efficiency factorstrategy that an efficiency factor is introduced in model forprioritizing the constrained terms over the objective term

Data test shows that compared with the path of boomlength changing before optimization both the number of


Cylinder


(a) Full retraction


0 045 091





I II III IV V

(c) Full extension











StepNO

Non-optimized

Drivensection

Optimization1

Drivensection

Optimization2

Drivensection

Optimization3

Drivensection













(0)

Startpoint

-045

-045

0

-045

0

000 0

+045

(045) (045) (045) (09) (045)

(045)(045)(045)(0)(045)

1

2

1

2

2

2A1 A2 A3

V1 V2 V3 1

2A4

V4 2

2A5

V5

2T51T42T31T22T1

(0) (0) (045) (045) (09)

-045

-0450 0 -0

+045

+045

+045

+045

+0


119878119887119900119900119898 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111989511988910038161003816100381610038161003816) (3)

119878119888119910119897119894119899119889119890119903 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111990011989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111990011989511988910038161003816100381610038161003816) (4)






119864 = 1198602 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1198612 119898sum

119894=1

( 119899sum119909=1

119881119909119894 minus 1)2+ 1198632 119899sum119909=1 119899sum119910=1 119898sum119894=1119881119909119894119889119909119910119881119909119894119881119910119894+1

(5)




119881 =(((

1 0 0 01 0 0 00 1 0 01 0 0 00 1 0 0)))

119881 times 119889 =(((

000450045)))

(6)



119864TPO = 1198612 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1205822

sdot 119899sum119896=1

[S119896( 119896sum119909=1

119898sum119894=1


[[(119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894)2

+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119894119889119894)2+ ( 119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)2+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119900119894119889119894)2]]119872119896 = max( 119898sum

119894=1

119860119896119894119889119894 119898sum119894=1

119879119896119894119889119894) k = 1 119899g119896 (119881) = [ 119896sum

119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1]

119878119896 = 0 g119896 le 01 g119896 gt 0

(7)






1198922 = ( 2sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +1198722 minus 1) le 0 (9)


119892119899 = ( 119899sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119899 minus 1) le 0 (10)





119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)


Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)


(14)









119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)



100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018









Cylinder


(a) Full retraction


0 045 091





I II III IV V

(c) Full extension











StepNO

Non-optimized

Drivensection

Optimization1

Drivensection

Optimization2

Drivensection

Optimization3

Drivensection













(0)

Startpoint

-045

-045

0

-045

0

000 0

+045

(045) (045) (045) (09) (045)

(045)(045)(045)(0)(045)

1

2

1

2

2

2A1 A2 A3

V1 V2 V3 1

2A4

V4 2

2A5

V5

2T51T42T31T22T1

(0) (0) (045) (045) (09)

-045

-0450 0 -0

+045

+045

+045

+045

+0


119878119887119900119900119898 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111989511988910038161003816100381610038161003816) (3)

119878119888119910119897119894119899119889119890119903 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111990011989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111990011989511988910038161003816100381610038161003816) (4)






119864 = 1198602 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1198612 119898sum

119894=1

( 119899sum119909=1

119881119909119894 minus 1)2+ 1198632 119899sum119909=1 119899sum119910=1 119898sum119894=1119881119909119894119889119909119910119881119909119894119881119910119894+1

(5)




119881 =(((

1 0 0 01 0 0 00 1 0 01 0 0 00 1 0 0)))

119881 times 119889 =(((

000450045)))

(6)



119864TPO = 1198612 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1205822

sdot 119899sum119896=1

[S119896( 119896sum119909=1

119898sum119894=1


[[(119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894)2

+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119894119889119894)2+ ( 119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)2+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119900119894119889119894)2]]119872119896 = max( 119898sum

119894=1

119860119896119894119889119894 119898sum119894=1

119879119896119894119889119894) k = 1 119899g119896 (119881) = [ 119896sum

119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1]

119878119896 = 0 g119896 le 01 g119896 gt 0

(7)






1198922 = ( 2sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +1198722 minus 1) le 0 (9)


119892119899 = ( 119899sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119899 minus 1) le 0 (10)





119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)


Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)


(14)









119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)



100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018










StepNO

Non-optimized

Drivensection

Optimization1

Drivensection

Optimization2

Drivensection

Optimization3

Drivensection













(0)

Startpoint

-045

-045

0

-045

0

000 0

+045

(045) (045) (045) (09) (045)

(045)(045)(045)(0)(045)

1

2

1

2

2

2A1 A2 A3

V1 V2 V3 1

2A4

V4 2

2A5

V5

2T51T42T31T22T1

(0) (0) (045) (045) (09)

-045

-0450 0 -0

+045

+045

+045

+045

+0


119878119887119900119900119898 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111989511988910038161003816100381610038161003816) (3)

119878119888119910119897119894119899119889119890119903 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111990011989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111990011989511988910038161003816100381610038161003816) (4)






119864 = 1198602 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1198612 119898sum

119894=1

( 119899sum119909=1

119881119909119894 minus 1)2+ 1198632 119899sum119909=1 119899sum119910=1 119898sum119894=1119881119909119894119889119909119910119881119909119894119881119910119894+1

(5)




119881 =(((

1 0 0 01 0 0 00 1 0 01 0 0 00 1 0 0)))

119881 times 119889 =(((

000450045)))

(6)



119864TPO = 1198612 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1205822

sdot 119899sum119896=1

[S119896( 119896sum119909=1

119898sum119894=1


[[(119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894)2

+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119894119889119894)2+ ( 119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)2+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119900119894119889119894)2]]119872119896 = max( 119898sum

119894=1

119860119896119894119889119894 119898sum119894=1

119879119896119894119889119894) k = 1 119899g119896 (119881) = [ 119896sum

119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1]

119878119896 = 0 g119896 le 01 g119896 gt 0

(7)






1198922 = ( 2sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +1198722 minus 1) le 0 (9)


119892119899 = ( 119899sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119899 minus 1) le 0 (10)





119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)


Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)


(14)









119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)



100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018









(0)

Startpoint

-045

-045

0

-045

0

000 0

+045

(045) (045) (045) (09) (045)

(045)(045)(045)(0)(045)

1

2

1

2

2

2A1 A2 A3

V1 V2 V3 1

2A4

V4 2

2A5

V5

2T51T42T31T22T1

(0) (0) (045) (045) (09)

-045

-0450 0 -0

+045

+045

+045

+045

+0


119878119887119900119900119898 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111989511988910038161003816100381610038161003816) (3)

119878119888119910119897119894119899119889119890119903 = 5sum119895=1

(10038161003816100381610038161003816119860119895119889 minus 11988111990011989511988910038161003816100381610038161003816 + 10038161003816100381610038161003816119879119895119889 minus 11988111990011989511988910038161003816100381610038161003816) (4)






119864 = 1198602 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1198612 119898sum

119894=1

( 119899sum119909=1

119881119909119894 minus 1)2+ 1198632 119899sum119909=1 119899sum119910=1 119898sum119894=1119881119909119894119889119909119910119881119909119894119881119910119894+1

(5)




119881 =(((

1 0 0 01 0 0 00 1 0 01 0 0 00 1 0 0)))

119881 times 119889 =(((

000450045)))

(6)



119864TPO = 1198612 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1205822

sdot 119899sum119896=1

[S119896( 119896sum119909=1

119898sum119894=1


[[(119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894)2

+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119894119889119894)2+ ( 119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)2+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119900119894119889119894)2]]119872119896 = max( 119898sum

119894=1

119860119896119894119889119894 119898sum119894=1

119879119896119894119889119894) k = 1 119899g119896 (119881) = [ 119896sum

119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1]

119878119896 = 0 g119896 le 01 g119896 gt 0

(7)






1198922 = ( 2sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +1198722 minus 1) le 0 (9)


119892119899 = ( 119899sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119899 minus 1) le 0 (10)





119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)


Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)


(14)









119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)



100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018










119881 =(((

1 0 0 01 0 0 00 1 0 01 0 0 00 1 0 0)))

119881 times 119889 =(((

000450045)))

(6)



119864TPO = 1198612 119899sum119909=1( 119898sum119894=1119881119909119894 minus 1)2 + 1205822

sdot 119899sum119896=1

[S119896( 119896sum119909=1

119898sum119894=1


[[(119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894)2

+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119894119889119894)2+ ( 119898sum119894=1

119860119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)2+ ( 119898sum119894=1

119879119909119894119889119894119909

minus 119898sum119894=1

119881119909119900119894119889119894)2]]119872119896 = max( 119898sum

119894=1

119860119896119894119889119894 119898sum119894=1

119879119896119894119889119894) k = 1 119899g119896 (119881) = [ 119896sum

119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1]

119878119896 = 0 g119896 le 01 g119896 gt 0

(7)






1198922 = ( 2sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +1198722 minus 1) le 0 (9)


119892119899 = ( 119899sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119899 minus 1) le 0 (10)





119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)


Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)


(14)









119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)



100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018










119881119894 = 119891119894 (119880119894) = 12 (1 + 119886 tanh(119880119894119879 )) (11)


Δ119880 = 119889119880119909119894119889119905 = minus 120597119864120597119881119909119894 (12)

Δ119880 = minus119861( 119898sum119894=1

119881119909119894 minus 1)minus 120582 119899sum119896=1

119878119896( 119896sum119909=1

119898sum119894=1

119881(119909minus1)119894119889119894 +119872119896 minus 1)1198891198941015840+ 2120574119888( 119898sum

119894=1

119860119909119894119889119894+ 119898sum119894=1

119879119909119894119889119894 minus 119898sum119894=1

119881119909119894119889119894 minus 119898sum119894=1

119881119909119900119894119889119894)1198891198941015840(13)


(14)









119892119896 (119905) + 119870119889120582sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (16)

120582 (119905 + 1) = 120582 (119905) + 120583120582Δ120582 (17)


119892119896 (119905) + 119870119889120574sdot (119892119896 (119905) minus 119892119896 (119905 minus 1)) (18)



100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018









100

90

80

70

60

50

40

30

20

10

00 1000 2000 3000 4000 5000

112233

4455





minus1 0 1 2

00

05

10

＝Ｅ

＝Ｅ

ＡＥ



119888 = 119899prod119896=1

119888119896 (21)











Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

0

05

1

15

2

25

Tele

scop

ic B

oom

Len

gth

(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018














Section4Section5

A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 2 1 1 1]

0

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Len

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(a) V=[1 2 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 1 1 1]

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(b) V=[1 1 1 1 1]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[1 1 2 1 2]

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(c) 119881lowast=[1 1 2 1 2]


Section4Section5



A=[2 2 2 2 2]T=[2 1 2 1 2]V=[2 1 1 1 2]

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(d) 119881lowast=[2 1 1 1 2]








valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018














valid time


10 29 22 30 14 1245 95 65 70 42 48















Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018









Sgg1g2

g3g4g5

0 1000 2000 3000 4000 5000minus5

0

5



0 1000 2000 3000 4000 5000minus5

0

5


Sgg1g2

g3g4g5











05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018









05000

01024

t

V1(1)

Vmax1

Vind

1

050001

012

Vind

2

V2(1)

tVmax2

050001

012

Vind

3

V3(1)

tVmax30

50000

11

152

t

V4(1)

Vmax4Vi

nd4

050001

123

Vind

5

V5(2)

tVmax5

05 0608

0608 05

0608


05000

01024

t

V1(1)

Vmax1

Vind

1

05000

012

Vind

2

V2(1)

tVmax2

05000

012

Vind

3

V3(1)

tVmax30

50001123

Vind

4

V4(2)

tVmax4

050001

123

Vind

5

V5(2)

tVmax5

05 070809

0708090608

0608




Tele

scop

ic B

oom

Len

gth


A 13312T 22221RBC1 2111


Section4Section5

0

05

1

15

2

25



Tele

scop

ic B

oom

Len

gth


A 3312 1T 22221RBC1 2211


Section4Section5

0

05

1

15

2

25









PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018










PIDPID+c

[1 1 1 1 2] [1 1 1 2 2]

0

200

400

600

Ener

gy F

unct

ion 800

1000

1200

1000 2000 3000 4000 50000Iteration


0 500 1000 1500 20000

20

40

60

80

100

120

140

160


Iteration

Ener

gy F

unct

ion

HNNHNN+c

[1 2 2 1 2] [1 1 1 2 2]
















Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018













Data Availability




References


































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








hopfield neural network method for problem of telescoping

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