an optimized collision avoidance decision-making system

Research ArticleAn Optimized Collision Avoidance Decision-Making System forAutonomous Ships under Human-MachineCooperation Situations

XiaolieWu 1KezhongLiu12JinfenZhang3ZhitaoYuan12 JiongjiongLiu3andQingYu 1

1School of Navigation Wuhan University of Technology Wuhan 430063 China2Hubei Key Laboratory of Inland Shipping Technology Wuhan 430063 China3National Engineering Research Center for Water Transport Safety Wuhan University of Technology Wuhan 430070 China

Correspondence should be addressed to Qing Yu qingyuwhuteducn

Received 13 July 2021 Accepted 18 August 2021 Published 27 August 2021

Academic Editor Xinqiang Chen

Copyright copy 2021 Xiaolie Wu et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Maritime Autonomous Surface Ships (MASSs) are attracting increasing attention in recent years as it brings new opportunities forwater transportation Previous studies aim to propose fully autonomous system on collision avoidance decisions and operationseither focus on supporting conflict detection or providing with collision avoidance decisions However the human-machinecooperation is essential in practice at the first stage of automation An optimized collision avoidance decision-making system isproposed in this paper which involves risk appetite (RA) as the orientation)e RA oriented collision avoidance decision-makingsystem (RA-CADMS) is developed based on human-machine interaction during ship collision avoidance while being consistentwith the International Regulations for Preventing Collisions at Sea (COLREGS) and Ordinary Practice of Seamen (OPS) Itfacilitates automatic collision avoidance and safeguards the MASS remote control Moreover the proposed RA-CADMS are usedin several encounter situations to demonstrate the preference )e results show that the RA-CADMS is capable of providingaccurate collision avoidance decisions while ensuring efficiency of MASS maneuvering under different RA

1 Introduction

)e idea of embracing autonomous vessels to create sus-tainable development and shape new opportunities for watertransportations in the industry is getting feasible As thetechnological barriers are resolved in numerous studiesautonomous vessels have been drawing significant attentionrecently For autonomous vessels the collision avoidancesystem (CAS) and the control system are two fundamentalsystems which are closely related with vessel maneuveringand safety performance Both of them aim at removinghuman operator in the control loop which is still questionableas teaching the automation to understand regulations is stillunrealistic and the trust of public on the autonomous vesselsis challenging [1] However human intelligence and machineintelligence are complementary the former one is good atexperience and the later one shows high power of computing

[2] )erefore the human-machine cooperation is essentialfor developing automatic system for the Maritime Autono-mous Surface Ships (MASS) [3]

Several studies tried to propose intelligent collisionavoidance and ship control systems to support autonomousvessels navigating in complex waters However the majorityof them are purely based on geometrical collision avoidanceapproaches (eg CPA approach fuzzy logic [4] artificialpotential field method [5] velocity obstacle [6 7] modelpredictive control (MPC) [8] and Convention on the In-ternational Regulations for Preventing Collisions at Sea(COLREGS) ignoring the valuable experience from crew-members in practices leading to unreasonable decisionmaking for the autonomous vessel in real encounter situ-ations )is question has been highlighted in the firstgeneration of autonomous vessels in which the unmannedand manned vessels exist when the response of autonomous

HindawiJournal of Advanced TransportationVolume 2021 Article ID 7537825 17 pageshttpsdoiorg10115520217537825

vessels to collision risk is still uncertain [9] Automation canimprove vessel safety from some perspectives by removingthe crews while creating other unpredictable risks

In the real practice the collision avoidance decisions forthe manned vessels are made by the officers who are onwatching (OOWs) Although these decisions should beconsistent with COLREGS the OOWs will make differentchoices based on their risk appetite (RA) Several relatedfactors should be considered by the OOWs such as the watercondition ship conditions and weather conditions)erefore to develop an intelligent CAS for autonomousvessels the OOWsrsquo experience of RA and maneuveringbehaviors should be fully considered to make the proposedsystem more practical and comprehensible under man-machine symbiosis condition

To resolve the question the contribution of this paper is topioneer an optimized collision avoidance decision-makingsystem for autonomous ships in complex encounter situationsinvolving RA as the orientation )e RA oriented collisionavoidance decision-making system (RA-CADMS) is developedbased on human-machine interaction during ship collisionavoidance consistent with COLREGS and Ordinary Practice ofSeamen (OPS) It is capable of providing the decisions accordingto the RA under different encounter situations and the oper-atorrsquos risk tolerance reflecting OOWsrsquo experience undercomplex water situation that lacks information so that realizinghuman-machine cooperated collision avoidance decision-making )e study firstly analyzes the ship collision avoidancedecision behavior under various ship encounter situations)en a multicriteria ship collision avoidance decision-makingoptimization system is developed to demonstrate the decisionpreference of OOWs who have different RA At last a prospecttheory (PT) is applied to balance the accepted risk thresholds inthe field of ship collision avoidance system )e main contri-bution of this paper can be highlighted as follows

(1) A RA-CADMS is proposed to incorporate hybridintelligence that combines the complementarystrength of experience from OOWs

(2) )e balance of collision risk tolerance and avoidancedecision efficiency is resolved by the RA-CADMS bypioneering PT in the collision avoidance decision-making

(3) )e decision space for the MASS in different en-counter situations is visualized and can be selectedwith different levels of RA

)e rest of the paper is organized as follows previousstudies related to risk appetite and ship collisions arereviewed in Section 2 In Section 3 an optimized collisionavoidance decision-making system is established and sev-eral case studies are implemented in Section 4 )e dis-cussion and conclusions are drawn in Sections 5 and 6

2 Literature Review

21 Collision Avoidance System As defined by [10] a gen-eralized CAS should at least contain collision risk assess-ment action decision-making and action execution

modules in which the decision selection is the core of theCAS as it is essential for safe navigation of MASS [11]

To develop a sufficient CAS to support collisionavoidance various methods are proposed in the previousstudies )ey are grouped into three perspectives Firststatistical analysis and numerical method are used tosupport collision avoidance such as encounter situation andstage discrimination quantitative model [12] with collisionparameters Reference [13] proposed the Personifying In-telligent Decision-making for Vessel Collision Avoidance(PIDVCA) algorithm which quantified the initial timing ofsteering rudder and the last time for steering rudder On thebasis of the COLREGS Woerner et al [14] and Chen et al[6] suggested a quantitative evaluation approach to analysethe collision risk )e second group covers the knowledge-based marine collision avoidance system to improve thestability and comprehensibility of the models [15] )is typeof CAS selects the optimal collision avoidance schemethrough various ship dynamic parameters eg course [16]relative speed [10] and ship trajectories [17] )e appliedapproaches in the CAS include fuzzy logic [4] and Bayesiannetwork [18] )e last group applies advanced artificialintelligence to deal with the problem and has made greatprogress on collision avoidance system formulation Forexample Shen et al [19] utilised the deep reinforcementlearning method to solve the collision avoidance problemunder multiship encounter situations in restricted watersHu et al [20] developed a multiobjective optimizationmodel for COLREGS-compliant path planning improvedcultural particle swarm was introduced into ship collisionavoidance decision [21] To support multiple ships anti-collision decision CAS based on multiagent was designedin which the agents formulate collision-free strategythrough information interaction [22] such as distributedalgorithm [23 24] and centralized algorithm [25] More-over big data processing techniques for ship AIS trajectoriesand video detection provided well support to identify shipencounter behaviors more accurately [26 27] or study thetraffic flows in water areas [28]

It can be found that the section of collision avoidance isgreatly enriched by previous studies However most ofthem ignore the impacts from human that include thecommon practices of seafarers good seamanship and risktolerance

22 Collision Avoidance System for MASS )e currentstudies related to MASS collision avoidance can be cate-gorized into two groups [9] One aims to developing the alertsystems for MASS to detect potential dangers of collisionusing collision risk index [29ndash31] ship domain [32] dan-gerous region in velocity-space [33] and probability ofcollision [34] risk prediction based on deep learning [35]etc Different from the manned ship objects of CAS forMASS are conflict detections which are difficult in theperception of collision situations [9] Although the expert-based methods and model-based methods are widely pre-sented to support the CAS design for MASS limitations onthese methods are noted (1) there are no common

2 Journal of Advanced Transportation

agreements on the boundary to clarify the safety and danger(2) the CAS hardly provide personalized alarm services forofficers with different risk attitudes

Another group concentrates on the automatic collision-free solutions for MASS while the interactions of the hu-man-machine cooperation encounter situations are notconsidered According to the current studies (eg Huang etal [9]) there are six kinds of collision-free resolutionmethods for MASS including rule-based method virtualvector method discretization of solutions continuous so-lutions replanning method and hybrid method )reeobjectives include (1) offering an optimalfeasible scheme(2) safety checking on scheme and (3) defining all the unsafesituations However some of the methods are difficult to beapplied in unmanned situations For instance Johansen et al[8] proposed a predictive control model to offer collisionavoidance by considering the maneuvering ability while itmight be difficult for MASS controller to understand theeffects of these forces and how to select a collision schemeBesides the optimal solution offered by CAS may not beaccepted by the MASS controller as the risk attitude forindividuals is inconsistent

)e maneuver decision preferences for manned vesselvary with several influencing factors [36ndash38] investigatingthe maneuver decision preferences in navigating ships toavoid a collision and to ensure safety for the MASSHowever relevant studies are rarely found in the currentliterature due to the lack of theoretical studies

23 Risk Appetite in Decision-Making RA can be defined asthe amount and type of risk that an individual is willing topursue or retain or the willingness to take on risky activitiesin pursuit of values [39] A high-risk appetite means high-risktolerance and hopes to give large benefits and vice versa Forexample 60 officers perceived different risk scores and per-formed various ship maneuverings based on scenario surveyexperiments under the same encounter situations [37] )eprospect theory (PT) was firstly proposed by [40] and wasthen updated to cumulative prospect theory (CPT) [41] andthe third generation prospect theory [42] )e theory com-bines behavioral science theory with multicriteria decision-making methods to reflect decision-makersrsquo psychologicalcharacteristics and attitudes of loss and gain As the PT hasgreat advantages in describing the psychological behaviorcharacteristics and loss of decision makers it has been widelyused in financial analysis transportation and other domainsthat involve multiattribute decision-making For instanceWang et al [43] considered the psychological behavior in theemergency decision-making and proposed a group emer-gency decision-making approach based on PT which wascapable of adjusting the importance of the model attributes-based expertsrsquo psychological behavior Gao et al [44] pro-posed a path selection model based on a CPT further Hjorthand Fosgerau [45] used risk preference parameters of travelerto trade-off traveling cost and identify the sensitivity of eachparameter impacting the daily cost Moreover a multi-attribute decision-making method [46] based on PT wasproposed to solve risk decision-making problems with

interval probability Specifically Zhou et al [47] applied PTin route traffic analysis and studied the driversrsquo route choicebehavior and developed a route selection model to aid trafficmanagement

It can be found from the review that PT is a sufficientapproach to support human behavior-based risk modellingIt is suitable for developing intelligent CAS for the MASSthat crewmembersrsquo behavioral characteristics and riskpreferences can be covered to support collision avoidanceBesides the following problems need to be resolved duringthe collision avoidance decision-making

(i) How can a CAS combine the wisdom of human andintelligence of machine in collision avoidancedecision

(ii) How to balance the safety and economy of collisionavoidance decision and adjust to the crew individualrisk attitude variation in decision-making

3 A Generic Framework of RA-CADMS

An abstract representation of RA-CADMS is shown inFigure 1 It contains two subcomponents the RA assessorand the CADM model )e RA assessor confirms thepossible avoidance schemes for the current encounter sit-uations through three RA-related features of safety effi-ciency and stationarity )e CADM model assigns decisionweights to evaluate the features and then convert them intocomposite prospect value which ranks the collisionavoidance decisions for the MASS under individual riskattitude Moreover a real-time updated loop is added to linkthe two components It provides with self-updating for theRA-CADMS when information for MASS and target ships isupdated after the executed decision

31 RA Assessor A high-risk appetite means high-risk tol-erance and hopes to give large benefits and vice versa Forexample a survey implemented by [37] invites 60 officers tomake an avoidance decision under the same encountersituation Although the states of the vessel weather andother conditions are the same the selected starting point foravoidance andminimum distance of approaching for each ofthem are different as their RA andmaneuvering behavior aredifferent thereby developing a RA assessor based on OOWsrsquoOPS and COLREG is the major component in the RA-CADMS in which influence factors for collision avoidancethat related to RA are discussed

311 Parameter Related to Ship Collision Avoidance AsRule 16 of COLREG states that ldquoEvery vessel is directed tokeep out of the way of another vessel shall so far as possibletake early and substantial action to keep well clearrdquo the give-way vessel can take action at any time within the time in-terval between the time to urgent situation and the time toclose-quarter situation However there are great differencesin trajectory characteristics under the same waters based onhigh-quality ship trajectory data from AIS [27 48] Furthervideo-based detection reflected more detailed and clear ship

Journal of Advanced Transportation 3

encounter behavior such as moving straight turning rightand turning left in specific waters [49] )erefore a high RAdecision-maker may take avoidance action later than thetime interval and pass the encounter at a closer distance tosave fuel and time Contrary to the high one an OOW withlow RA is cautious and prefers to take action earlier tomaintain a safer distance so that leading to larger deviationand higher fuel consumptions On such basis three cate-gories of RA are stated as cautious (ie risk aversion) ag-gressive (ie risk hobbies) and neutral (ie risk neutrality)Cautious refers to low risk appetite and is willing to obtainhigher safety values aggressive means higher tolerance andneutral stands intermedium of RA

To describe the risk of encounter situations ship dy-namic parameters are selected in the RA assessor )eseinclude ship relative speed (vij) accepted distance (dacc)and change of course (θ) )us the action time t can beexpressed as t sim (vi vj dacc θ |Pos PTs) where POS and PTSare the positions for the own ship (OS) and the target ship

(TS) Assuming there is a MASS (ie ship i) encounteredwith a manned vessel (ie ship j) leads to an encountersituation of crossing over (see Figure 2) )e real speeds forthe ship i and the ship j are vi and vj respectively and therelative speed and position between two encounters are vij

and Pij To avoid collision the ship i should take a coursechanging action at a certain time t and position Pij(t) toensure that it could pass the ship j at a safety distance If theship i takes action at time t1 and position Pij(t1) theminimum course change is θ1 Similarly the ship i shouldchange course to θ2 when it gets closer (ie take action attime t2 and position Pij(t2))

312 Indicators for RA Assessor Based on COLREGS OPSand relevant researches [38 50 51] the optimization ofcollision avoidance decision-making for MASS is mainlyreflected in safety efficiency and stationarity )e threeindicators are proposed as follows

(1) Safety it is to ensure that the collision risk can beeliminated by the avoidance decisions Althoughships should pass each other not less than theminimum safety distance the accepted distancebetween two ships is not persistent in real practice Itis impacted by several factors such as weather shiptype and size and is selected based on OPS OOWsevaluate the safety value (SV) of decision schememainly by the minimum passing distance (dm) andthe time-taking evasive action to passing the closepoint time (TCPAa) )us we define SV asexpressed in the following equations

SV dm( 1113857

dm

d11113888 1113889

065

1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

dm le d1 dm led1

d1 ltdm d1 lt dm

(1)

SV TCPAa( 1113857

TCPAa

t1

1

⎧⎪⎪⎨

⎪⎪⎩

0leTCPAa le t1

TCPAa lt 0 or t1 ltTCPAa(2)

SV w1SV dm( 1113857 + w2SV TCPAa( 1113857 dm leD1 and 0leTCPAa le t1

1 others1113896 (3)

)e thresholds D1 and t1 are defined to divide thesafety value SV with two segments respectivelyAssuming a MASS minimum passing another TSmore than d1 or navigation at a range larger than t1no collision risk is existed (ie SV 1)

(2) Efficiency this indicator evaluates the distance de-viating (Dev) of the OS from its initial course LargeDev stands for low efficiency and small Dev means

high efficiency It can be calculated using the fol-lowing equation

Dev v times t2 minus t2( 1113857 times sin θ (4)

)e Dev is influenced by the OSprime speed v coursedeviation θ and the time between the course recovertime t2 and the course change time t1 A sample isgiven in Figure 3

RAassessor

indicators value

indicators selection

parameter analysis

TS and OS information

composite prospectvalue

prospect value anddecision weight

reference points

decision matrix andnormalized matrix

scheme generation

CADMmodel

Figure 1 )e framework for RA-CADMS


(3) Stationarity this indicator describes the smoothnessand stationarity (Sta) of the shiprsquos trajectory in col-lision avoidance operation According to Rule 18 ofCOLREGS ship course andor speed alteration shouldbe apparent to another ship observing visually or byradar )erefore the course change θ of decisionschemes is at least 20deg At the same time consideringship maneuvering dynamics model and existing re-searches [52 53] abrupt and too large course changecould cause ship speed slowing down and bunkerconsumption )is indicator function is as follows

Sta π minus |θ| (5)

32 CADM Model )is section proposes a CADM modelbased on PT to sort the decision for different risk appetiteofficers )e advantage is that it takes into account the

psychological characteristics through risk parameters fordecision makers in the uncertainty decision progress In theCADM model the collision avoidance schemes are priori-tized based on composite prospect values which can becalculated through the equations introduced in the PT Asshown in Figure 4 the calculation mainly follows two stepsFirst the indicator value for each scheme that selected in theRA assessor is normalized to construct the decision matrixwhile the reference points for the indicators are identified byusing the grey correlation analysis Second the prospect andobjective values are aggregated by using PT which considerthe RA of the MASS and the weight of the indicators

321 Prospect 3eory )e PT was first proposed by Kah-neman and Tversky [54] to explain the human behavior in asystematic way It formulates the biased or irrational humanbehaviors and introduces two concepts of the definition ofprospect a value function defined on the utility and a de-cision weight function defined on the cumulative proba-bility as shown in Figure 5

As shown in Figure 5(a) the value function is convexwhen the curve is in the area of gains and is concave when inthe losses area )e curve in the value function is divided bythe reference points ξminus

ij for the positive prospect value and ξ+ij

for the negative prospect value it is steeper for losses thanfor gains and can be formulated with prospect valueequations

)e positive prospect value (ie area of gains) for ithscheme and jth indicator is calculated as

pv+ij 1 minus ξminus

ij1113872 1113873α (6)

and the negative prospect value (ie area of losses) for ithscheme and jth indicator is calculated as

pvminusij minus η times minus ξ+

ij minus 11113872 11138731113960 1113961β (7)

where α β and η are the risk preference parameters )eincreasing value of the parameters means the model is moresensitive on loss so that trends to avoid risk

Figure 5(b) reports how the decision weight functionsdescribe the well-observed behavior that human tends tooverestimate the occurrence of low-probability events butunderestimate that of high probability ones as

π(ω)

π(ω)+

ωc

ωc+(1 minus ω)

c1113858 1113859

1c ξminusj is reference point

π(ω)minus

ωδ

ωτ+(1 minus ω)

τ1113858 1113859

1τ ξ+j is reference point

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(8)

where ω represents the objective probability of the eventresult c and τ represent the fitting parameters of theprobability weight function on the left and right sides of thereference point

Composite prospect value of each scheme is defined asfollows

θ

t1 course-change time

t2 course-recover time

doc

Figure 3 A sample of collision avoidance decision

x

y

o

Ship i

vi

vij (t)

vij (t1)vij (t2)

Pij (t1)

Pij (t2)

dacc

θ1

θ1

θ2

θ2

Ship j

vj

Figure 2 A sample of the autonomous vessel (own ship) en-countered with a manned vessel (target ship)


pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (9)

322 Decision Matrix To develop the CADM model adecision matrix should be established to describe thenumber of n available collision avoidance schemesU1 U2 Un1113864 1113865 For instance under an encounter situationshown in Figure 6 there are three available collisionavoidance schemes U1 U2 U31113864 1113865 )e first schemeU1 (t1 P1 θ1 dacc1) means when the OS is moving to theposition P1 at time t1 the collision risk is unacceptable for aMASS with the RA of caution )e MASS makes a coursechange θ1 and finally passes the TS with an accepted distanceof dacc1 (green line in Figure 6) Similarly the avoidanceschemes for neutral RA and aggressive RA areU2 (t2 P2 θ2 d2) and U3 (t3 P3 θ3 d3) respectivelyleading to accepted distance of dacc2 (yellow line in Figure 6)and dacc3 (red line in Figure 6)

Each U contains the number of m decision-makingindicators x1 x2 xm1113864 1113865 )e decision-making matrix Xcan be represented as follows

X

x11 x12 middot middot middot x1m


⋮ ⋮ ⋱ ⋮

xn1 xn2 middot middot middot xnm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

In the decision matrix each indicator (x1 SVx2 Dev and x3 Sta) that is obtained from RA assessorneeds to be normalized into an interval of [minus 1 1] to acquire anormalized decisionmatrixR by using linear transformationequations in which

Zj 1n

1113944

n

i1xij (11)

For indicator of Dev

rij zj minus xij

max maxj

xij1113872 1113873 minus zj zj minus minj

xij1113872 11138731113896 1113897

(12)

For indicator of Sv and Sta

v (u)

gains

losses

0 u

(a)

00

10

10

w (p)

p

(b)

Figure 5 An example of (a) the value function and (b) the weighting function

Prospect value matrix

Composite prospectvaluegr

ey co

rrel

atio

n an

alys

is

Decision schemes andindicators

Decision matrix andnormalization

correlation coefficientmatrix

reference points

Decision indicatorsweight

Risk parameters ofOOWs

pros

pect

theo

ry

Figure 4 )e framework for CMDA model


rij xij

minus zj

max maxj


xij1113872 11138731113896 1113897

(13)

323 Reference Points It can be noted that the used ref-erence points are crucial for evaluating the performance ofthe indicators in a PT-based model For a CADMmodel theselections of the reference points are associated with OPSOnce the reference points are selected the optional decisionscheme can be evaluated by yielding prospect values betweenthe positive (ie gains) and negative (ie losses) In otherwords the selection of reference points in the CADMmodelmeans the expression of OOWsrsquo knowledge and experience

As each OOW has hisher own experience to determinethe gains and the losses under different encounter situationssetting a global value for universal scenarios is not rea-sonable )us the CADM model applies a grey correlationanalysis technology to determining the reference pointunder different situations )e technology is based on themethod of proximity measure of similarity which cancalculate the degree of association between factors and usethe parameter to indicate the reference point )e correla-tion coefficient for each indicator under different collisionavoidance schemes can be calculated as follows

ξ+ij

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

ξminusij

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

(14)

where r+j max rij|1le ile n1113966 1113967 rminus

j min rij|1le ile n1113966 1113967 j

1 2 m and ρ isin [0 1] while positive and negativereference points for each indicator are identified as follows

ξ+j max ξ+

ij|1le ile n1113966 1113967 j 1 2 m

ξminusj max ξminus

ij|1le ile n1113966 1113967 j 1 2 m(15)

324 Composite Prospect Value Based on PT the com-posite prospect value of each scheme can be calculated byusing the following equation

pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (16)

and the aggregated prospect value can be calculated by usingthe objective function

max pv 1113944m

i11113944

n

j1pv+

ijπ+ ωj1113872 1113873 + 1113944

m

i11113944

n

j1pvminus

ijπminus ωj1113872 1113873 (17)

where the weight ωlowast (ωlowast1 ωlowast2 ωlowastm) for each indicatoris obtained based on the userrsquos RA and OPS in which

st0leωj le 11113936j

ωj 1⎧⎨

⎩ It can be assigned by expert judgements and

linear programming

4 Case Study

To validate the proposed system the simulations are carriedout to test the effectiveness of the RA-CADM model )esimulation test contains two typical ship encounter sce-narios which are carried out in MATLAB software using thei7-8550U CPU and 12GB RAM In the simulations one ofthe scenarios describes a MASS encounters an individualmanned vessel and another presents a multiple encountersituation

41 ShipManoeuvrability Simulation )e Nomoto model isused to simulate the ship movement when the ships take acollisions avoidance action To simplify the model theimpacts from wind and current are not taken into consid-eration Consequently the ship course changes are simu-lated by using the following equation

T times _r + r K times δ (18)

where T is the constant time for a ship changing its courseand K is the rudder gain r is the angular speed and δ is therudder angle )e course change after time of t can becalculated as

θ K times δ0 times t minus T + T times eminus (tT)

1113872 1113873 (19)

while the K can be calculated with K VL and T 2 times LVbased on experience

x

y

oShip i

vi

vij (t)

vij (t1)

vij (t2)

Pij (t1)

Pij (t2)

Pij (t3)

vij (t3)θ1

θ2

θ3

θ

Ship jvj

Figure 6 An example of ship encounter situation


42 Scenario I Pairwise Vessel Encounter Based on thesimulations the first target is to test the typical individualcollision avoidance scenario As shown in Table 1 the initialencounter scenario is that the OS (ie MASS) is headingnorth along the planned route while the TS comes from thestarboard side with bearing of 049deg According to COLREGSthe MASS should perform its duty of giving way to thecoming ship )en the simulation assumes that both shipsare equipped with AIS which is able to broadcast shipdynamic information with a period of 10 seconds when aship remains its course or with a period of 3 seconds whenthe ship is taking a course change

)e initial information for both ships is given in Table 1)espeeds forOS andTS are 11 and 13 knots respectively the initialcourse for the MASS is 000deg and the initial course for the TS is275deg )e relative distance between the two ships is 7 nauticalmiles and the lengths for them are 225meters and 189meters Inaddition two vessels are encountered at the open and calm sea

To avoid collision theMASS tries to keep clear of all the TSunder six possible accepted distances where dacc isin (10 12

14 16 18 20) In terms of the course changes six degreesare provided for the MASS from low to highθ isin (20deg 25deg 30deg 35deg 40deg 45deg) which reflect the RA ofOOWs In order to test the collision avoidance behavior moreapparently Table 2 offers the details on the collision avoidanceschemes including the accepted distance course of changesaction times recover time and initial route deviation

As the examples the OSprime trajectory for scheme of U11 (the

blue trajectory) and U16 (the green trajectory) and TSrsquos tra-

jectory (pink) are demonstrated in Figure 7 In the scheme ofU1

1 theMASS decides to pass the TSwith an accepted distanceof 10 nm thus it needs to change the course to 020deg at theaction time of 121 minutes After it passed the TS the MASSrecovers its course to 000deg at 264minutes)e whole collisionavoidance action takes 143 minutes In the scheme of U1

6 theMASS changes its course to 045deg at 88 minutes to pass the TSwith an acceptable distance of 20 nm After passing the TStheMASS recovers its course to 000deg at 243 minutes)e totalaction time for scheme U6 is 155 minutes

)e RA-CADM model is used to evaluate the collisionavoidance schemes that are given in Table 3 and the mostsuitable scheme is selected for the MASS under different riskpreferences (ie caution neutral and aggressive) )eoverall process includes five steps taking aggressive as anexample (Tversky 1992) (η 225 α β 088 c 061

τ 069)

Step 1 calculating the RA indicator for each scheme)e values of RA indicator for the scheme can becalculated using equations (1)ndash(5) Using the schemeU1 as an example the RA values of safety efficiencyand stationarity can be calculated as SVU1 08times

(1020)065 + 02 times (13525) 0618 DevU1 11times

264 minus 12160 times sin 20deg 0897 and StaU1 π minus 20degtimes π180deg 2793 where d1 10 t1 25 TCPAU1

256 and TCPAU1 256 minus 121 135 Similarly thevalues of the RA indicators for the other 5 schemes arecalculated to construct the decision-making matrix Xwhich is

X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

Step 2 normalization )e RA indicator values in the Xare then normalized to acquire the normalized decisionmatrix R in which equations (11)ndash(13) are used tocalculate the normalized RA values for safety effi-ciency and stationarity respectively As a result

R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

Step 3 identifying the reference points To evaluate theperformance of the RA indicators in the RA-CADMmodel the maximum and minimum values of eachindicator are selected from the normalized decisionmatrix R )e positive set (ie the maximum value set)is r+ 0903 1 1 and negative set (ie the minimumvalue set) is rminus minus 1 minus 0801 minus 1 Using equations (14)and (15) the r11 in the R is divided to two correlationcoefficient values ξ+

11 and ξminus11 where

ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 minus 0903| + 05 times 2 0345

ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)

Table 1 Scenario I encounter information

OS TSSOG (knots) 11 13COG 000deg 275degDistance (nm) 7 7Relative bearing (deg) mdash 049Ship length (meters) 225 189


Table 2 Scenario I collision avoidance schemes

No of schemes Accepted distance (nm) Change ofcourse (degrees) Action time (minutes) Recover time (minutes) Initial route deviation (nm)

U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20

Table 3 Scenario II initial encounter information

OS TS1 TS2SOG (knots) 14 10 10COG 000deg 210deg 225degDistance (nm) mdash 6 8Relative bearing (deg) mdash 027 021DCPA (nm) mdash 142 050

t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)

Figure 7 Ship trajectories in Case 1


Similarly the normalized decision matrix R is con-verted to the positive correlation coefficient matrix

ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)

and the negative correlation coefficient matrix

ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ

minusij the maximum and the minimum values in

each column are selected as the reference points tocalculate the positive prospect value pv+ and negativeprospect value pvminus by using equations (6) and (7) Inthe calculation the risk preference parameters aredefined as an aggressive RA that is α β 088

)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and

pvminus11 minus 225 times (1 minus 0345)

088 minus 1552


088 0


088 0

⎧⎪⎪⎨

⎪⎪⎩ As results

the positive prospect matrix

pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and the negative prospect

matrix pvminus

minus 1552 0 0minus 1420 minus 0809 minus 0747minus 1250 minus 1161 minus 1102minus 1014 minus 1357 minus 1320minus 0657 minus 1456 minus 1468minus 0546 minus 1525 minus 1575

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are

obtainedStep 4 assigning weights to the indicators With aidsof a linear interactive optimize software program(LINGO 150) the optimal weight assignments ωlowast

(ω1ω2 ω3) for each RA indicator can be simulatedTo set the maximum target function as MAX (0 +

0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +

(1 minus w1)061)1061 + (minus 1552 minus 142 minus 125 minus 1014minus

0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069

while defining the limitation of outputs as

st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

the weight assignments are generated

as ωlowast 05 04 01 Step 5 calculating the composite prospect value for eachcollision avoidance scheme )e composite prospectvalues for each scheme are calculated by aggregating theabove information using equation (18) )ereby pvU1

minus 025 pvU2 minus 053 pvU3 minus 065 pvU4 minus 070 pvU5 minus 069 and pvU6 minus 086 Based on the results theoptimized ranking for an aggressive RA user

Aggressive sim U11 ≻U

12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)

Similarly if the RA for the MASS is set as neutral theparameter set of η 225 α β 099 and c τ 099should be used according to previous studies [55] )eoptimum collision avoidance schemes are resequenced as

Neutral sim U14 ≻U

15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)

where the composite prospect values for the schemes arepvU1 minus 067 pvU2 minus 066 pvU3 minus 062 pvU4 minus 050

pvU5 minus 055 and pvU6 minus 073Meantime for RA is cautious the parameter value as-

signment is η 35 α 121 β 102 c 055 and τ

049 [56] )e composite prospect values are calculated aspvU1 minus 125 pvU2 minus 122 pvU3 minus 120 pvU4 minus 110

pvU5 minus 089 andpvU6 minus 087 )e collision avoidanceschemes are ranked as

Cautious sim U16 ≻U

15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)

43 Scenario II Multiple-Vessel Encounter As the proposedRA-CADM model is able to deal with multiple-vessel en-counter situations the second scenario simulates an en-counter situation that involves 2 TSs (as shown in Figure 8)and the ship details are given in Table 3

As shown in Figure 8 two TSs come from starboard ofthe MASS in which the TS1 will keep clear of the MASS andthe TS2 will cross the OSprime safety distance and lead to highrisk of collision To avoid the urgent risk of collision theMASS needs to select an optimal collision scheme that notonly keeps clear of the TSs but also ensures the efficiency ofthe action As stated in the COLREGS that a give-way shipshall change course to starboard to avoid collision therebyonly two collision avoidance strategies can be considered bythe MASS (1) changing course to starboard with largedegree to keep clear the TSs but suffering bigger deviationdistance (ie low efficiency) or (2) remaining its coursebefore passing the TS1 and then turning starboard to avoidthe TS2 so that the deviation distance will be relatively smallOn such a basis the available collision avoidance schemesare shown in Table 4 in which the former three schemes U2

1


U22 and U2

3 are the smaller deviation strategy and thesubsequent three U2

4 U25 and U2

6 are the large deviationstrategy

)e simulation results of the TS1 (pink) TS2 (black) andMASSrsquos trajectories by applying the schemes U2

1 U21 (blue)

and U26 U2

6 U26 (green) are given in Figure 9

By using the same step introduced above the compositeprospect value for each collision avoidance scheme in thesecond scenario can be obtained that pvU1 minus 025 pvU2

minus 044 pvU3 minus 053 pvU4 minus 071 pvU5 minus 070 andpvU6 minus 087 )erefore the collision avoidance schemesfor aggressive RA are ranked as


22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)

Similarly the collision avoidance schemes of neutral andcautious officers are ranked as


22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)

5 Results and Discussion

51 Indicator Values under Different RA )e case studyresults show that the RA-CADMS is capable of aiding theMASS to select the optimal collision avoidance schemesbased on the RA and risk tolerance To validate the ratio-nality of the outputs from the model this section discussesthe variation of the RA indicators in terms of a dynamicprospective

511 RAAggressive As shown in Figure 10 when settingthe RA for a MASS to avoid collision as aggressive theschemes of U1

1 and U21 are chosen as the optimal action for

the MASS under scenarios I and II respectively In thescenario I the dynamic values of three RA indicators arereported in Figure 10(a) It can be noted the MASS takescourse change at time 725 s when the SV is relatively low(04) After the action is made the SV for the MASS isincreasing to 06 but the Dev began to grow and reached10 nm in the end In scenario II (see Figure 10(b)) theMASS takes actions at time 845 s and the SV is increasedfrom 105 to 120 causing the deviation of 09 nm

512 RANeutral When RA is set as neutral the RA-CADMS selects the schemes U1

4 and U23 as two optimal deci-

sions for the MASS under scenario I and scenario II As shownin Figure 11(a) the MASS chooses to take collision avoidanceaction at time 560 s when SV is 043 As a result the SV sig-nificantly raised to 079 and theDev increased to 17 In scenarioII (see Figure 11(b) theMASS takes action at time 835 s and theSV is increased from 105 to 125 leading to a Dev of 17nm

513 RACautious Similarly when RA cautious theoptimal schemes for theMASS are U1

6 (for scenario I) and U24

(for scenario II) As shown in Figure 12(a) in scenario I theaction time for the MASS is 520 s when SV is 045 after thecollision avoidance action is took the SV increases tomaximum but the Dev is also relatively large (Dev 20) In

Table 4 Scenario II collision avoidance scheme

Scheme Accepted distance (n mile) Change of course (degree) Action time (minutes) Recover time (minutes)U2

1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2

DCPA1=142 nmDCPA2=050 nm

y [n

mile

]5

5

0

RML1

RML2

x [n mile]

TCPA1=150 minTCPA2=215 min

Figure 8 )e multiple-vessel encounter situation


scenario II Figure 12(b) theMASS chooses to avoid two TSsat the initial time of the simulation For that moment the SVis 16 )e action causes a large route deviation which is thelargest among all the schemes (Dev 5)

52 Characteristics of Different RA )e simulation resultsdemonstrate that different RA setting in the MASS leads tothe different collision avoidance actions proving a closerelationship among them To characterize the avoidanceactions under different RA can provide better understandingfor MASS designercontrollers

Table 5 reports the details of the scheme selections Fivefactors are considered including the action time SV valuesbefore action SV values after action and SV increment anddeviation Based on Table 4 the following findings areconcluded

(1) For MASS the more conservative it is the easier it isto take collision avoidance action earlier It is evidentthat the cautious RA selects the earliest action time(520 seconds) in scenario I which is 205 secondsearlier than the scheme that is selected by the ag-gressive RA )is phenomenon is more apparent inscenario II as the scheme U2

6 takes action at initialtime while the U2

1 takes action at 845 seconds)is isdue to the risk tolerance under different RA Forinstance in scenario I the minimum accepted SV forthe aggressive RA is 04 while the minimum ac-cepted SV for cautious is 045 )e high tolerance ofrisk causes later actions

(2) Followed by the first findings this study notes thatthe safety levels under different RA are different Forinstance in the scheme U1

1 of scenario I the SV raisesto maximum (ie SV 1) after the collision

t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)



avoidance action is taken by a cautious RA with asignificant increasing on SV (from 045 to 055) Incontrast the MASS with high-risk tolerance (ieaggressive) accepts SV 06 as the safety level Itmeans the action taken by aggregative RA is riskier

(3) Under different risk appetites the composite pros-pect value of the same collision avoidance schemesfor the same encountering scenario is different )ecomposite prospect value of the aggressive crew islarger than that of the cautious crew for the

aggressive pursuing greater prospect value but theprospect value of each scheme is negative in any typeof officers indicating that there exists expected de-viation compared to ideal decision

(4) In practice MASSs can recommend the officersoptimal schemes according to the specific risk ap-petite types of different operators which can notonly ensure the safety and economy of ship collisionavoidance action but also be well understood andaccepted by the OOWs

RA=aggressive

recover time=1585 saction time=725 s

20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive

action time=845 s recover time=1415 s

500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)

Figure 10 Results of decision indicators with aggressive officers in (a) scenario I and (b) scenario II

RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral

action time=835 srecover time=1405 s

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)

StaDistance-TS1Distance-TS2

DevSV

(b)

Figure 11 Results of decision indicators with neutral officers in (a) scenario I and (b) scenario II


(5) )e relationships between cost (ie deviation)and profit (ie SV) are not linear By simulating100 cases of increasing the deviation from 01 nm

to 10 nm the relationship between the deviationand the SV can be fitted with the followingequation

X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6

Figure 13 )e fitness between the deviation and the increment of the SV

RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)

Figure 12 Results of decision indicators with cautious officers in (a) scenario I and (b) scenario II

Table 5 Results under different RA

RA Collision avoidance scheme Action time (second)SV

DeviationBefore action After action Increment

Scenario IAggressive U1

1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2

Scenario IIAggressive U2

1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)

As shown in Figure 13 the fitness function indicates thatbefore deviation less than 15 nm a slight avoidance actionwould increase the SV significantly which is a good ratio ofbenefitWhen the deviation is larger than 15 nm the ratio ofbenefit for ships that takes a large change is unsatisfactory

6 Conclusion and Future Research

)is study proposed a RA-CADMmodel to support collisionavoidance decision making for MASS under man-machinehybrid conditions On the basis of the collision risk modelthe novelty of the model included the following (1) it firstlyapplied the prospect theory in collision avoidance decisionmaking and pioneered the MASS control with a human-likemachine system (2) the proposed model was tested withscenario simulations that covered the individual collisionscenario and multiple collision scenarios to prove the reli-ability and the applicability in real cases (3) the RA underdifferent states were discussed in-depth so as to presentsufficient support for MASS designer and controller to makecollision avoidance strategies Although the results of thiscase study demonstrate risk preference these risk param-eters from other researches may deviate from the actualcrews due to industry background differences

)erefore further research works are suggested as follows(1) the differences between traditional CAS and intelligent CAScan be discussed (2) the weight assignment for the RA in-dicators in the model should be investigated through surveysand or AIS data mining (3) considering more risk-relatedfactors during the decision-making process to optimize thecollision schemes and (4) real ship tests are encouraged forobtaining more information of collision avoidance decision

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (NSFC) under Grant No 52031009

References

[1] Z Liu Y Zhang X Yu and C Yuan ldquoUnmanned surfacevehicles an overview of developments and challengesrdquo An-nual Reviews in Control vol 41 pp 1ndash23 2016

[2] Y Huang L Chen P Chen R R Negenborn andP H A J M Van Gelder ldquoShip collision avoidance methodsstate-of-the-artrdquo Safety Science vol 121 pp 451ndash473 2019

[3] P Koopman and M Wagner ldquoAutonomous vehicle safety aninterdisciplinary challengerdquo IEEE Intelligent TransportationSystems Magazine vol 9 no 1 pp 90ndash96 2017

[4] L P Perera J P Carvalho and C Guedes Soares ldquoFuzzy logicbased decisionmaking system for collision avoidance of oceannavigation under critical collision conditionsrdquo Journal ofMarine Science and Technology vol 16 no 1 pp 84ndash99 2011

[5] H Lyu and Y Yin ldquoCOLREGS-constrained real-time pathplanning for autonomous ships using modified artificialpotential fieldsrdquo Journal of Navigation vol 72 no 3pp 588ndash608 2019

[6] P Chen Y Huang J Mou and P H A J M Van GelderldquoProbabilistic risk analysis for ship-ship collision state-of-the-artrdquo Safety Science vol 117 pp 108ndash122 2018

[7] Y Huang L Chen and P H A J M Van Gelder ldquoGen-eralized velocity obstacle algorithm for preventing ship col-lisions at seardquo Ocean Engineering vol 173 pp 142ndash156 2018

[8] T A Johansen T Perez and A Cristofaro ldquoShip collisionavoidance and COLREGS compliance using simulation-basedcontrol behavior selection with predictive hazard assessmentrdquoIEEE Transactions on Intelligent Transportation Systemsvol 17 no 12 pp 3407ndash3422 2016

[9] Y Huang L Chen R R Negenborn andP H A J M Van Gelder ldquoA ship collision avoidance systemfor human-machine cooperation during collision avoidancerdquoOcean Engineering vol 217 Article ID 107913 2020

[10] L P Perera J P Carvalho and C Guedes Soares ldquoIntelligentocean navigation and fuzzy-Bayesian decisionaction for-mulationrdquo IEEE Journal of Oceanic Engineering vol 37 no 2pp 204ndash219 2012

[11] L P Perera J P Carvalho and C G Soares ldquoSolutions to thefailures and limitations of Mamdani fuzzy inference in shipnavigationrdquo IEEE Transactions on Vehicular Technologyvol 63 no 4 pp 1539ndash1554 2013

[12] Y He Y Jin L Huang Y Xiong P Chen and J MouldquoQuantitative analysis of COLREG rules and seamanship forautonomous collision avoidance at open seardquo Ocean Engi-neering vol 140 no April pp 281ndash291 2017

[13] L Li G Chen and G Li ldquoStudy on auto decision-making andits simulation control for vessel collision avoidancerdquo Pro-ceedings of the 2015 Chinese Intelligent Automation Confer-ence vol 338 pp 265ndash275 2015

[14] K Woerner M R Benjamin M Novitzky and J J LeonardldquoQuantifying protocol evaluation for autonomous collisionavoidancerdquo Autonomous Robots vol 43 no 4 pp 967ndash9912019

[15] Q Yu K Liu C-H Chang and Z Yang ldquoRealising advancedrisk assessment of vessel traffic flows near offshore windfarmsrdquo Reliability Engineering amp System Safety vol 203Article ID 107086 2020

[16] W Naeem G W Irwin and A Yang ldquoCOLREGs-basedcollision avoidance strategies for unmanned surface vehiclesrdquoMechatronics vol 22 no 6 pp 669ndash678 2012

[17] M-C Fang K-Y Tsai and C-C Fang ldquoA simplified sim-ulation model of ship navigation for safety and collisionavoidance in heavy traffic areasrdquo Journal of Navigationvol 71 no 4 pp 837ndash860 2018

[18] Q Yu K Liu Z Yang H Wang and Z Yang ldquoGeometricalrisk evaluation of the collisions between ships and offshoreinstallations using rule-based Bayesian reasoningrdquo ReliabilityEngineering amp System Safety vol 210 Article ID 107474 2021

[19] H Shen H Hashimoto A Matsuda Y Taniguchi D Teradaand C Guo ldquoAutomatic collision avoidance of multiple ships


based on deep Q-learningrdquo Applied Ocean Research vol 86pp 268ndash288 2018

[20] L Hu W Naeem E Rajabally et al ldquoA multiobjective op-timization approach for COLREGs-Compliant path planningof autonomous surface vehicles verified on networked bridgesimulatorsrdquo IEEE Transactions on Intelligent TransportationSystems vol 21 no 3 pp 1167ndash1179 2020

[21] Y Zheng X Zhang Z Shang S Guo and Y Du ldquoA decision-making method for ship collision avoidance based on im-proved cultural particle swarmrdquo Journal of AdvancedTransportation vol 2021 Article ID 8898507 31 pages 2021

[22] S Li J Liu and R R Negenborn ldquoDistributed coordinationfor collision avoidance of multiple ships considering shipmaneuverabilityrdquo Ocean Engineering vol 181 no 1178pp 212ndash226 2019

[23] D Kim K Hirayama and T Okimoto ldquoDistributed sto-chastic search algorithm for multi-ship encounter situationsrdquoJournal of Navigation vol 70 no 4 pp 699ndash718 2017

[24] J Zhang D Zhang X Yan S Haugen and C Guedes SoaresldquoA distributed anti-collision decision support formulation inmulti-ship encounter situations under COLREGsrdquo OceanEngineering vol 105 pp 336ndash348 2015

[25] R Szlapczynski ldquoEvolutionary sets of safe ship trajectorieswithin traffic separation schemesrdquo Journal of Navigationvol 66 no 1 pp 65ndash81 2013

[26] X Chen Z Li Y Yang L Qi and R Ke ldquoHigh-resolutionvehicle trajectory extraction and denoising from aerialvideosrdquo IEEE Transactions on Intelligent TransportationSystems vol 22 no 5 pp 3190ndash3202 2020

[27] M Liang R W Liu S Li Z Xiao X Liu and F Lu ldquoAnunsupervised learning method with convolutional auto-en-coder for vessel trajectory similarity computationrdquo OceanEngineering vol 225 Article ID 108803 2021

[28] Q Yu K Liu A P Teixeira and C G Soares ldquoAssessment ofthe influence of offshore wind farms on ship traffic flow basedon AIS datardquo Journal of Navigation vol 73 no 1 pp 131ndash1482020

[29] W Zhang F Goerlandt J Montewka and P Kujala ldquoAmethod for detecting possible near miss ship collisions fromAIS datardquo Ocean Engineering vol 107 pp 60ndash69 2015

[30] Y Wen Y Huang C Zhou J Yang C Xiao and X WuldquoModelling of marine traffic flow complexityrdquo Ocean Engi-neering vol 104 pp 500ndash510 2015

[31] Z Cheng Y Li and BWu ldquoEarly warningmethod andmodelof inland ship collision risk based on coordinated collision-avoidance actionsrdquo Journal of Advanced Transportationvol 2020 Article ID 5271794 14 pages 2020

[32] R Szlapczynski and J Szlapczynska ldquoReview of ship safetydomains models and applicationsrdquo Ocean Engineeringvol 145 pp 277ndash289 2017

[33] Y Huang and P H A J M Van Gelder ldquoTime-varying riskmeasurement for ship collision preventionrdquo Risk Analysisvol 40 no 1 pp 24ndash42 2020

[34] J Park M Han and W Baek ldquoQuantifying the performanceimpact of large pages on in-memory big-data workloadsrdquo inProceeding 2016 IEEE International Symposium on WorkloadCharacterization IISWC Providence RI US May 2016

[35] J Ma W Li C Jia C Zhang and Y Zhang ldquoRisk predictionfor ship encounter situation awareness using long short-termmemory based deep learning on intership behaviorsrdquo Journalof Advanced Transportation vol 2020 Article ID 889770015 pages 2020

[36] B Lin ldquoBehavior of ship officers in maneuvering to prevent acollisionrdquo Journal of Marine Science and Technology vol 14no 4 pp 225ndash230 2006

[37] J-B Yim D-S Kim and D-J Park ldquoModeling perceivedcollision risk in vessel encounter situationsrdquo Ocean Engi-neering vol 166 pp 64ndash75 2018

[38] C D Wickens A Williams B A Clegg and C A P SmithldquoNautical collision avoidancerdquoHuman Factors 3e Journal ofthe Human Factors and Ergonomics Society vol 62 no 8pp 1304ndash1321 2020

[39] T Aven ldquoOn the meaning and use of the risk appetiteconceptrdquo Risk Analysis vol 33 no 3 pp 462ndash468 2013

[40] D Kai-Ineman and A Tversky ldquoProspect theory an analysisof decision under riskrdquo Econometrica vol 47 no 2pp 363ndash391 1979

[41] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

[42] U Schmidt C Starmer and R Sugden ldquo)ird-generationprospect theoryrdquo Journal of Risk and Uncertainty vol 36no 3 pp 203ndash223 2008

[43] L Wang Y-M Wang and L Martınez ldquoA group decisionmethod based on prospect theory for emergency situationsrdquoInformation Sciences vol 418-419 pp 119ndash135 2017

[44] S Gao E Frejinger and M Ben-Akiva ldquoAdaptive routechoices in risky traffic networks a prospect theory approachrdquoTransportation Research Part C Emerging Technologiesvol 18 no 5 pp 727ndash740 2010

[45] K Hjorth and M Fosgerau ldquoUsing prospect theory to in-vestigate the low marginal value of travel time for small timechangesrdquo Transportation Research Part B Methodologicalvol 46 no 8 pp 917ndash932 2012

[46] P Liu F Jin X Zhang Y Su and M Wang ldquoResearch on themulti-attribute decision-making under risk with intervalprobability based on prospect theory and the uncertain lin-guistic variablesrdquo Knowledge-Based Systems vol 24 no 4pp 554ndash561 2011

[47] L Zhou S Zhong S Ma and N Jia ldquoProspect theory basedestimation of driversrsquo risk attitudes in route choice behaviorsrdquoAccident Analysis amp Prevention vol 73 pp 1ndash11 2014

[48] R W Liu J Nie S Garg Z Xiong Y Zhang andM S Hossain ldquoData-driven trajectory quality improvementfor promoting intelligent vessel traffic services in 6G-enabledmaritime IoT systemsrdquo IEEE Internet 3ings Journal vol 8no 7 pp 5374ndash5385 2020

[49] X Chen L Qi Y Yang et al ldquoVideo-based detection in-frastructure enhancement for automated ship recognition andbehavior analysisrdquo Journal of Advanced Transportationvol 2020 Article ID 7194342 12 pages 2020

[50] J Li H Wang Z Guan and C Pan ldquoDistributed multi-objective algorithm for preventing multi-ship collisions atseardquo Journal of Navigation vol 73 no 5 pp 971ndash990 2020

[51] Y Hu A Zhang W Tian J Zhang and Z Hou ldquoMulti-shipcollision avoidance decision-making based on collision riskindexrdquo Journal of Marine Science and Engineering vol 8no 9 p 640 2020

[52] K Zhou J Chen and X Liu ldquoOptimal collision-avoidancemanoeuvres to minimise bunker consumption under the two-ship crossing situationrdquo Journal of Navigation vol 71 no 1pp 151ndash168 2018

[53] L Hu W Naeem E Rajabally et al ldquoCOLREGs-compliantpath planning for autonomous surface vehicles a multi-objective optimization approach lowast lowast the authors should liketo thank innovate UK grant reference tsb 102308 for the


funding of this projectrdquo IFAC-PapersOnLine vol 50 no 1pp 13662ndash13667 2017

[54] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2pp 263ndash292 1979

[55] H Xu J Zhou and W Xu ldquoA decision-making rule formodeling travelersrsquo route choice behavior based on cumu-lative prospect theoryrdquo Transportation Research Part CEmerging Technologies vol 19 no 2 pp 218ndash228 2011

[56] Z Jian-min ldquoAn experimental test on cumulative prospecttheoryrdquo University Natural Sciences Education vol 1 2007


vessels to collision risk is still uncertain [9] Automation canimprove vessel safety from some perspectives by removingthe crews while creating other unpredictable risks

In the real practice the collision avoidance decisions forthe manned vessels are made by the officers who are onwatching (OOWs) Although these decisions should beconsistent with COLREGS the OOWs will make differentchoices based on their risk appetite (RA) Several relatedfactors should be considered by the OOWs such as the watercondition ship conditions and weather conditions)erefore to develop an intelligent CAS for autonomousvessels the OOWsrsquo experience of RA and maneuveringbehaviors should be fully considered to make the proposedsystem more practical and comprehensible under man-machine symbiosis condition

To resolve the question the contribution of this paper is topioneer an optimized collision avoidance decision-makingsystem for autonomous ships in complex encounter situationsinvolving RA as the orientation )e RA oriented collisionavoidance decision-making system (RA-CADMS) is developedbased on human-machine interaction during ship collisionavoidance consistent with COLREGS and Ordinary Practice ofSeamen (OPS) It is capable of providing the decisions accordingto the RA under different encounter situations and the oper-atorrsquos risk tolerance reflecting OOWsrsquo experience undercomplex water situation that lacks information so that realizinghuman-machine cooperated collision avoidance decision-making )e study firstly analyzes the ship collision avoidancedecision behavior under various ship encounter situations)en a multicriteria ship collision avoidance decision-makingoptimization system is developed to demonstrate the decisionpreference of OOWs who have different RA At last a prospecttheory (PT) is applied to balance the accepted risk thresholds inthe field of ship collision avoidance system )e main contri-bution of this paper can be highlighted as follows

(1) A RA-CADMS is proposed to incorporate hybridintelligence that combines the complementarystrength of experience from OOWs

(2) )e balance of collision risk tolerance and avoidancedecision efficiency is resolved by the RA-CADMS bypioneering PT in the collision avoidance decision-making

(3) )e decision space for the MASS in different en-counter situations is visualized and can be selectedwith different levels of RA

)e rest of the paper is organized as follows previousstudies related to risk appetite and ship collisions arereviewed in Section 2 In Section 3 an optimized collisionavoidance decision-making system is established and sev-eral case studies are implemented in Section 4 )e dis-cussion and conclusions are drawn in Sections 5 and 6

2 Literature Review

21 Collision Avoidance System As defined by [10] a gen-eralized CAS should at least contain collision risk assess-ment action decision-making and action execution

modules in which the decision selection is the core of theCAS as it is essential for safe navigation of MASS [11]

To develop a sufficient CAS to support collisionavoidance various methods are proposed in the previousstudies )ey are grouped into three perspectives Firststatistical analysis and numerical method are used tosupport collision avoidance such as encounter situation andstage discrimination quantitative model [12] with collisionparameters Reference [13] proposed the Personifying In-telligent Decision-making for Vessel Collision Avoidance(PIDVCA) algorithm which quantified the initial timing ofsteering rudder and the last time for steering rudder On thebasis of the COLREGS Woerner et al [14] and Chen et al[6] suggested a quantitative evaluation approach to analysethe collision risk )e second group covers the knowledge-based marine collision avoidance system to improve thestability and comprehensibility of the models [15] )is typeof CAS selects the optimal collision avoidance schemethrough various ship dynamic parameters eg course [16]relative speed [10] and ship trajectories [17] )e appliedapproaches in the CAS include fuzzy logic [4] and Bayesiannetwork [18] )e last group applies advanced artificialintelligence to deal with the problem and has made greatprogress on collision avoidance system formulation Forexample Shen et al [19] utilised the deep reinforcementlearning method to solve the collision avoidance problemunder multiship encounter situations in restricted watersHu et al [20] developed a multiobjective optimizationmodel for COLREGS-compliant path planning improvedcultural particle swarm was introduced into ship collisionavoidance decision [21] To support multiple ships anti-collision decision CAS based on multiagent was designedin which the agents formulate collision-free strategythrough information interaction [22] such as distributedalgorithm [23 24] and centralized algorithm [25] More-over big data processing techniques for ship AIS trajectoriesand video detection provided well support to identify shipencounter behaviors more accurately [26 27] or study thetraffic flows in water areas [28]

It can be found that the section of collision avoidance isgreatly enriched by previous studies However most ofthem ignore the impacts from human that include thecommon practices of seafarers good seamanship and risktolerance

22 Collision Avoidance System for MASS )e currentstudies related to MASS collision avoidance can be cate-gorized into two groups [9] One aims to developing the alertsystems for MASS to detect potential dangers of collisionusing collision risk index [29ndash31] ship domain [32] dan-gerous region in velocity-space [33] and probability ofcollision [34] risk prediction based on deep learning [35]etc Different from the manned ship objects of CAS forMASS are conflict detections which are difficult in theperception of collision situations [9] Although the expert-based methods and model-based methods are widely pre-sented to support the CAS design for MASS limitations onthese methods are noted (1) there are no common





















SV dm( 1113857

dm

d11113888 1113889

065

1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

dm le d1 dm led1

d1 ltdm d1 lt dm

(1)

SV TCPAa( 1113857

TCPAa

t1

1

⎧⎪⎪⎨

⎪⎪⎩

0leTCPAa le t1



1 others1113896 (3)






RAassessor

indicators value


parameter analysis




reference points


scheme generation

CADMmodel













ij1113872 1113873α (6)



ij minus 11113872 11138731113960 1113961β (7)



π(ω)

π(ω)+

ωc

ωc+(1 minus ω)

c1113858 1113859


π(ω)minus

ωδ

ωτ+(1 minus ω)

τ1113858 1113859


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(8)



θ



doc


x

y

o

Ship i

vi

vij (t)

vij (t1)vij (t2)

Pij (t1)

Pij (t2)

dacc

θ1

θ1

θ2

θ2

Ship j

vj



pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (9)



X



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)


Zj 1n

1113944

n

i1xij (11)


rij zj minus xij

max maxj


xij1113872 11138731113896 1113897

(12)


v (u)

gains

losses

0 u

(a)

00

10

10

w (p)

p

(b)




ey co

rrel

atio

n an

alys

is




reference points



pros

pect

theo

ry



rij xij

minus zj

max maxj


xij1113872 11138731113896 1113897

(13)



ξ+ij

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

ξminusij

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

(14)


j min rij|1le ile n1113966 1113967 j


ξ+j max ξ+

ij|1le ile n1113966 1113967 j 1 2 m


ij|1le ile n1113966 1113967 j 1 2 m(15)


pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (16)


max pv 1113944m

i11113944

n

j1pv+

ijπ+ ωj1113872 1113873 + 1113944

m

i11113944

n

j1pvminus

ijπminus ωj1113872 1113873 (17)



ωj 1⎧⎨


linear programming

4 Case Study






1113872 1113873 (19)


x

y

oShip i

vi

vij (t)

vij (t1)

vij (t2)

Pij (t1)

Pij (t2)

Pij (t3)

vij (t3)θ1

θ2

θ3

θ

Ship jvj













τ 069)


(1020)065 + 02 times (13525) 0618 DevU1 11times



X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)



ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2


ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)






U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References

















































































SV dm( 1113857

dm

d11113888 1113889

065

1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

dm le d1 dm led1

d1 ltdm d1 lt dm

(1)

SV TCPAa( 1113857

TCPAa

t1

1

⎧⎪⎪⎨

⎪⎪⎩

0leTCPAa le t1



1 others1113896 (3)






RAassessor

indicators value


parameter analysis




reference points


scheme generation

CADMmodel













ij1113872 1113873α (6)



ij minus 11113872 11138731113960 1113961β (7)



π(ω)

π(ω)+

ωc

ωc+(1 minus ω)

c1113858 1113859


π(ω)minus

ωδ

ωτ+(1 minus ω)

τ1113858 1113859


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(8)



θ



doc


x

y

o

Ship i

vi

vij (t)

vij (t1)vij (t2)

Pij (t1)

Pij (t2)

dacc

θ1

θ1

θ2

θ2

Ship j

vj



pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (9)



X



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)


Zj 1n

1113944

n

i1xij (11)


rij zj minus xij

max maxj


xij1113872 11138731113896 1113897

(12)


v (u)

gains

losses

0 u

(a)

00

10

10

w (p)

p

(b)




ey co

rrel

atio

n an

alys

is




reference points



pros

pect

theo

ry



rij xij

minus zj

max maxj


xij1113872 11138731113896 1113897

(13)



ξ+ij

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

ξminusij

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

(14)


j min rij|1le ile n1113966 1113967 j


ξ+j max ξ+

ij|1le ile n1113966 1113967 j 1 2 m


ij|1le ile n1113966 1113967 j 1 2 m(15)


pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (16)


max pv 1113944m

i11113944

n

j1pv+

ijπ+ ωj1113872 1113873 + 1113944

m

i11113944

n

j1pvminus

ijπminus ωj1113872 1113873 (17)



ωj 1⎧⎨


linear programming

4 Case Study






1113872 1113873 (19)


x

y

oShip i

vi

vij (t)

vij (t1)

vij (t2)

Pij (t1)

Pij (t2)

Pij (t3)

vij (t3)θ1

θ2

θ3

θ

Ship jvj













τ 069)


(1020)065 + 02 times (13525) 0618 DevU1 11times



X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)



ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2


ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)






U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References








































































ij1113872 1113873α (6)



ij minus 11113872 11138731113960 1113961β (7)



π(ω)

π(ω)+

ωc

ωc+(1 minus ω)

c1113858 1113859


π(ω)minus

ωδ

ωτ+(1 minus ω)

τ1113858 1113859


⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(8)



θ



doc


x

y

o

Ship i

vi

vij (t)

vij (t1)vij (t2)

Pij (t1)

Pij (t2)

dacc

θ1

θ1

θ2

θ2

Ship j

vj



pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (9)



X



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)


Zj 1n

1113944

n

i1xij (11)


rij zj minus xij

max maxj


xij1113872 11138731113896 1113897

(12)


v (u)

gains

losses

0 u

(a)

00

10

10

w (p)

p

(b)




ey co

rrel

atio

n an

alys

is




reference points



pros

pect

theo

ry



rij xij

minus zj

max maxj


xij1113872 11138731113896 1113897

(13)



ξ+ij

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

ξminusij

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

(14)


j min rij|1le ile n1113966 1113967 j


ξ+j max ξ+

ij|1le ile n1113966 1113967 j 1 2 m


ij|1le ile n1113966 1113967 j 1 2 m(15)


pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (16)


max pv 1113944m

i11113944

n

j1pv+

ijπ+ ωj1113872 1113873 + 1113944

m

i11113944

n

j1pvminus

ijπminus ωj1113872 1113873 (17)



ωj 1⎧⎨


linear programming

4 Case Study






1113872 1113873 (19)


x

y

oShip i

vi

vij (t)

vij (t1)

vij (t2)

Pij (t1)

Pij (t2)

Pij (t3)

vij (t3)θ1

θ2

θ3

θ

Ship jvj













τ 069)


(1020)065 + 02 times (13525) 0618 DevU1 11times



X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)



ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2


ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)






U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References






























































pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (9)



X



⋮ ⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)


Zj 1n

1113944

n

i1xij (11)


rij zj minus xij

max maxj


xij1113872 11138731113896 1113897

(12)


v (u)

gains

losses

0 u

(a)

00

10

10

w (p)

p

(b)




ey co

rrel

atio

n an

alys

is




reference points



pros

pect

theo

ry



rij xij

minus zj

max maxj


xij1113872 11138731113896 1113897

(13)



ξ+ij

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

ξminusij

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

(14)


j min rij|1le ile n1113966 1113967 j


ξ+j max ξ+

ij|1le ile n1113966 1113967 j 1 2 m


ij|1le ile n1113966 1113967 j 1 2 m(15)


pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (16)


max pv 1113944m

i11113944

n

j1pv+

ijπ+ ωj1113872 1113873 + 1113944

m

i11113944

n

j1pvminus

ijπminus ωj1113872 1113873 (17)



ωj 1⎧⎨


linear programming

4 Case Study






1113872 1113873 (19)


x

y

oShip i

vi

vij (t)

vij (t1)

vij (t2)

Pij (t1)

Pij (t2)

Pij (t3)

vij (t3)θ1

θ2

θ3

θ

Ship jvj













τ 069)


(1020)065 + 02 times (13525) 0618 DevU1 11times



X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)



ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2


ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)






U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References






























































rij xij

minus zj

max maxj


xij1113872 11138731113896 1113897

(13)



ξ+ij

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

ξminusij

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

(14)


j min rij|1le ile n1113966 1113967 j


ξ+j max ξ+

ij|1le ile n1113966 1113967 j 1 2 m


ij|1le ile n1113966 1113967 j 1 2 m(15)


pv 1113944ij

pv+ijπ

+ ωj1113872 1113873 + 1113944ij

pvminusijπ

minus ωj1113872 1113873 (16)


max pv 1113944m

i11113944

n

j1pv+

ijπ+ ωj1113872 1113873 + 1113944

m

i11113944

n

j1pvminus

ijπminus ωj1113872 1113873 (17)



ωj 1⎧⎨


linear programming

4 Case Study






1113872 1113873 (19)


x

y

oShip i

vi

vij (t)

vij (t1)

vij (t2)

Pij (t1)

Pij (t2)

Pij (t3)

vij (t3)θ1

θ2

θ3

θ

Ship jvj













τ 069)


(1020)065 + 02 times (13525) 0618 DevU1 11times



X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)



ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2


ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)






U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References








































































τ 069)


(1020)065 + 02 times (13525) 0618 DevU1 11times



X

0618 0897 2793

0692 1178 2705

0759 1448 2618

0822 1693 2531

0880 1862 2443

0934 2009 2356

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


R

r11 r12 r13

r21 r22 r23

r31 r32 r33

r41 r42 r43

r51 r52 r53

r61 r62 r63

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 1000 1000 1000

minus 0552 0545 0600

minus 0150 0107 0200

0225 minus 0289 minus 0200

0575 minus 0562 minus 0600

0903 minus 0801 minus 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)



ξ+11

mini

minj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus r+j

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2


ξminus11

mini

minj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868 + ρmaxi

maxj

rij minus rminusj

11138681113868111386811138681113868

11138681113868111386811138681113868

0 + 05 times 2

| minus 1 + 1| + 05 times 2 1

(22)






U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References
































































U11 10 20 121 264 09

U12 12 25 108 260 12

U13 14 30 100 258 15

U14 16 35 94 255 17

U15 18 40 90 248 18

U16 20 45 88 243 20



t=100 s

TS

OS (U11) OS (U6

1)

OS (U11)

OS (U61)

OS (U11)

OS (U11)

OS (U61)

OS (U61)

t=600 s

TS

t=800 s

TS

t=1680 s

TS

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7

y (n

m)

-2 -1 0 1 2 3 4 5 6-3x (nm)

-2

-1

0

1

2

3

4

5

6

7y

(nm

)




ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References































































ξ+ij

0345 1000 1000

0407 0687 0714

0487 0528 0556

0596 0437 0455

0753 0390 0385

0800 0357 0333

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)


ξminusij

1000 0357 0333

0691 0426 0385

0541 0524 0455

0450 0661 0556

0388 0807 0714

0345 1000 1000

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(24)

In ξ+ij and ξ



)erebypv+

11 (1 minus 1)088

0pv+

12 (1 minus 0357)088

0678pv+

13 (1 minus 0333)088

07

⎧⎪⎪⎨

⎪⎪⎩and


088 minus 1552


088 0


088 0

⎧⎪⎪⎨



pv+

0 0678 07000356 0613 06520504 0520 05870591 0386 04900649 0235 03320690 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


matrix pvminus


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

are



0356 + 0504 + 0591 + 0649 + 069) times (w0611 w061

1 +


0657 minus 0546) times (w0691 w069

1 + (1 minus w1)069 )1069


st

0leωj le 11113944

j

ωj 1

w1 ge 05

⎧⎪⎪⎪⎨

⎪⎪⎪⎩





12 ≻U

13 ≻U

15 ≻U

14 ≻U

161113872 1113873 (25)



15 ≻U

13 ≻U

12 ≻U

11 ≻U

161113872 1113873 (26)







15 ≻U

14 ≻U

13 ≻U

12 ≻U

111113872 1113873 (27)



1


U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References






























































U22 and U2


4 U25 and U2



1 U21 (blue)

and U26 U2





22 ≻U

23 ≻U

25 ≻U

24 ≻U

261113872 1113873 (28)



22 ≻U

24 ≻U

21 ≻U

25 ≻U

261113872 1113873


25 ≻U

24 ≻U

23 ≻U

22 ≻U

211113872 1113873

(29)














1 10 20 136 236U2

2 12 30 138 237U2

3 14 40 14 236U2

4 16 50 03 250U2

5 18 55 03 248U2

6 20 60 02 248

OS

TS1

TS2


y [n

mile

]5

5

0

RML1

RML2

x [n mile]












t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References






































































t=50 s

TS1

TS2

t=1000 s

TS1

TS2

t=1250 s

TS2

TS1

OS(U12)

OS(U12)

OS(U12)

OS(U12)

t=1650 s

TS2

TS1

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10

y (n

m)

-2 0 2 4 6 8-4x (nm)

-2

0

2

4

6

8

10y

(nm

)

OS(U62)

OS(U62)

OS(U62)

OS(U62)







RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References


































































RA=aggressive


20001000 1500 25000 500t (s)

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=aggressive


500 1500 2000 250010000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)


RA=neutral


25001500 200010005000t (s)

0

1

2

3

4

5

6

7

y (v

alue

)

SVDev

StaDistance

(a)

RA=neutral


0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

2000 2500500 150010000t (s)


DevSV

(b)





X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References
































































X 15Y 000562

times10-3

times10-3

35

4

45

5

55

6

65

7

75

Incr

emen

t of S

V

1 2 3 4 5 6 7 8 9 100Deviation (nm)

1 15 2 25 305

52545658

6


RA=cautious


15001000 25000 500 2000t (s)

0

1

2

3

4

5

6

7y

(val

ue)

SVDev

StaDistance

(a)

RA=cautious

action time=0s

recover time=1480s

2000 25001000 15005000t (s)

0

1

2

3

4

5

6

7

8

9

10

y (v

alue

)

DevStaDistance-TS1

Distance-TS2SV

(b)






1 725 04 06 02 1Neutral U1

4 560 043 079 036 17Cautious U1

6 520 045 1 055 2


1 845 105 12 015 09Neutral U2

3 835 105 125 02 18Cautious U2

6 0 16 2 04 5


f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References






























































f(Dev) 004739lowast dev113+ 01079 (30)





Data Availability




Acknowledgments


References






























































an optimized collision avoidance decision-making system

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