Collision risk detection and quantification in ship navigation with integrated bridge systems

Ocean Engineering 109 (2015) 344–354

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Collision risk detection and quantification in ship navigation with integrated bridge systems Lokukaluge P. Perera 1, C. Guedes Soares n Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

art ic l e i nf o

a b s t r a c t

Article history: Received 1 November 2014 Accepted 13 August 2015

This study focuses on a collision detection methodology and collision risk assessment in an integrated bridge system accounting for vessel state uncertainties in complex ship maneuvers. Modern technology solutions in integrated bridge systems (IBSs) to improve the navigation safety under vessel close encounters are discussed and ship navigation tools that would detect potential collision situations ahead of time are presented. The proposed vessel relative distance based collision risk detection and quantification methodology is simulated and evaluated under a two vessel encounter in a collision or near-collision situation. Furthermore, the relative navigation trajectory, the relative course-speed vector and the relative bearing vector of one vessel with respect to the other vessel are estimated by an extended Kalman filter. Then, the respective course-speed and relative bearing vectors are used to detect and quantify the collision risk between the vessels. Hence, the proposed collision risk detection and quantification methodology can be implemented in modem integrated bridge system (IBSs), where the potential risk among vessels ahead of a collision situation can be detected and that is also an important part of e-navigation. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Collision risk detection and quantification Ship collision avoidance e-Navigation Intelligent transport system Integrated bridge systems

1. Introduction 1.1. Navigation technology The estimation accuracy of position, velocity, acceleration and heading components of a vessel is an important part of a navigation system to predict its future positions and headings and to improve the safety of navigation. Historically, these navigation parameters are observed with respect to the sun, the moon, the stars and other environmental references (i.e. land mask, river banks, etc.). Modern technology has facilitated to achieve various complex navigation requirements (i.e. complex maneuvers) in land, air and in maritime transportation systems, where various navigation information sources (i.e. global and onboard) are integrated to obtain the required accuracy, integrity, availability and continuity conditions that are demanded. One should note that modern technological advances have been extensively implemented under various land (Skog and Handel, 2009) and air (Oosterom et al., 2002) transportation systems to satisfy the complex navigation requirements in the recent years.

Maritime transportation is still under developed with respect to these technological advances and a considerable number of navigation functionalities (i.e. propulsion control, engine control, and navigation prediction and collision avoidance) are still provided by human operators. These inadequate navigation functionalities can also compromise the safety of the respective operations under congested sea routes and complex ship maneuvers (i.e. harbor navigation, mine hunting, drilling, pipe-laying, replenishment at sea, diving support, dredging and dynamic positioning) under varying sea conditions. One should note that some port and offshore operations (i.e. port navigation, cable and pipe laying, offshore exploration and exploitation tugs operations, ice-breaking operations and narrow inland navigation) the navigation accuracy of 1 m to 2.5 m is required (Fairbanks et al., 2004). Even though the required safety levels in these operations can often be achieved by conventional methods of good ship-handling, seamanship training and safety procedures, it may have various limitations in complex vessel maneuvers under congested ship routes and/or varying sea conditions. 1.2. Ship maneuvers

n

Corresponding author. Tel.: þ 351 218417957. E-mail address: [email protected] (C. Guedes Soares). 1 Presently at the Norwegian Marine Technology Research Institute (MARINTEK), Trondheim, Norway. http://dx.doi.org/10.1016/j.oceaneng.2015.08.016 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

An experienced navigator needs to be placed on the bridge for a long period to satisfy these ship safety and operation requirements (i.e. to avoid potential hazard conditions of grounding, collision or

L.P. Perera, C. Guedes Soares / Ocean Engineering 109 (2015) 344–354

near collision and weather damaged conditions). However, the placement of an experienced navigator always on the bridge may not be a realistic approach, where the reported data show 75–96% of maritime accidents and causalities are caused by various human errors (Rothblum et al., 2002; Antão and Guedes Soares, 2008). Therefore, an experienced ship navigator can also make wrong decisions that resulted in human casualties and ship structural failures. Furthermore, these ship structural failures can result in various environmental disasters such as the vessels Tricolor and Kariba collision in 2002 near the north-south shipping route in the English Channel (Kerckhof et al., 2004), the Erika in 1999, the Prestige in 2002 (Eide et al., 2007), and the Costa Concordia in January 2012. Appropriate navigation decisions play an important role in ship collision and near collision situations and those decisions can eventually be transformed into collision avoidance actions with ship path planning approaches (Tam et al., 2009). Therefore, the respective course and speed changes (i.e. the collision avoidance actions) should be executed by the vessels to reduce the collision risk in these close encounters. However, these actions can also be influenced by the navigator’s intuition and his perception of the ship encounter situation, where the respective collision risk can also be erroneously quantified. In general, human behavior under extreme navigation situations (i.e. collision and near collision situations) can contribute to increase the respective collision risk. e.g. human intuition guides to reduce vessel/vehicle speed to avoid collision and near collision situations and that may not be applicable in many ship encounter situations, where the speed reduction in the vessel can also reduce its steering controllability. Furthermore, the respective navigation constraints and the routing schemes can further enforce vessels to execute close quarter maneuvers, where the respective collision risk is further increased (Robson, 2006). One should note that inadequate controllability, i.e. inadequate course change actions in a vessel can lead to a collision or near collision situation even under an experienced navigator. The respective collision risk in these ship encounters can further be increased by vessel-to-vessel interaction forces and moments (Sutulo et al., 2012), and navigation and communication instrument failures due the close proximity between vessels. That situation has often been categorized as the “Trapping Phenomenon” (Drouin and Bussieres, 2009) under close proximity navigation in confined waters (Zhou et al., 2014), which resulted under several reported maritime accidents such as the vessels in the Hyde Park and Cast Prosperity accident in 2005 in a dredged channel of the St. Lawrence river (Transportation Safety Board of Canada, 2005), the vessels in the Figaro and Camargue accident in 1988 at Galveston Bay, and the vessels of G. Ordzhonikidze and Lt. Argosy in 1996 at St. Lawrence river (Drouin and Bussieres, 2009). Therefore, appropriate ship speeds should be maintained to preserve adequate vessel controllability in close encounter situations to improve the navigation safety. One should note that additional measures can also be implemented by ship navigators to improve vessel controllability in some situations. i.e. in conventional ship navigation, reserve engine power should be available for improving vessel controllability in “kick ahead” situations (i.e. applying full rudder and utilizing short but substantial engine power) to overcome ship steering failures, instantly. However, these situations (i.e. ship steering failures) should be handled with extreme caution, while vessels are passing from deep to shallow water or vice versa. In general, the collision risk can be mitigated by executing a set of predefined navigation rules and regulations (IMO, 1972) as collision avoidance actions by the navigator ahead of a ship encounter situation (Perera et al., 2012c). The collision risk should be quantified appropriately in this situation, where the navigator

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has enough time to take the most suitable decisions (i.e. ship course and speed changes) to avoid the respective ship collision/ near collision situation (Statheros et al., 2008). This collision risk quantification process should be independent from an individual’s perception (i.e. individual experience) of the ship encounter situation, where a general set of navigation rules and regulations for collision avoidance among vessels should be derived (Perera et al., 2015a). Furthermore, the quantified collision risk information should also be distributed among the involved vessels to improve their awareness of the ship encounter situation. One should note that the improvements in human inference alone may not achieve the required navigation safety levels in these situations. Therefore, a distance based risk detection and quantification methodology for a potential ship collision situation with complex ship maneuvers is proposed in this study and that is the main contribution of the study.

2. e-Navigation 2.1. Global vision A global vision is needed to develop an international collaborative communication network to improve the safety in maritime transportation by integrating various navigation information sources (i.e. global and onboard). However, this collaborative network cannot be constructed from its foundation because various navigation platforms have already been implemented in the shipping industry. Hence, present navigation platforms should be adopted to achieve this vision of a collaborative network with traffic information among vessels and shore based authorities, which can enhance the navigation safety and improve the operating efficiency in shipping (Mitropoulos, 2006). This global vision is called “e-navigation” by the International Maritime Organization (IMO) and the International Association of Lighthouse Authorities (IALA). The integration among present navigation technologies and the implementation of intelligent decision support capabilities (Perera et al., 2012a; Nielsen and Jensen, 2011) with some limitations with human subjective factors in maritime transportation are proposed under this framework. One should note that e-navigation is classified as a user driven concept rather than a system driven concept at the present stage and that is formulated by IALA (IMO, 2007): The harmonized collection, integration, exchange, presentation and analysis of marine information onboard and ashore by electronic means to enhance berth to berth navigation and related services for safely and security at sea and protection of the marine environment. This can be seen as a strategic vision that consists of a global policy framework, which is established by the IMO subcommittees of Safety of Navigation (NAV) and Radio communications and Search and Rescue (COMSAR). Furthermore, the Norwegian Coastal Administration has also been entrusted to coordinate various related programs, initially. In addition, many independent research groups have also been working on many related topics of e-navigation to develop and implement existing and future navigation technologies (i.e. onboard, ashore and other communication elements) (IMO, 2013). However, this e-navigation vision will re-evaluate to eliminate the development conflicts and recover the missed opportunities in the technology advancements of ship navigation. The first major challenge in this process (i.e. e-navigation) is to integrate available navigation technologies in a proper format by considering the core navigation systems and sensors. The second major challenge is to invent new technologies that can support the

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vision of e-navigation. One should note that the concept of e-navigation is still in the deployment phase and that can also be influenced by future technological advancements. Hence, the e-navigation concept can eventually redefine “the navigator’s complete responsibility” in a voyage. The benefits under this concept can be elaborated as: increased navigation safety, collaborated vessel traffic situation awareness, improved traffic monitoring facilities and reduced shipping costs (Ward and Leighton, 2010). A pilot project in e-navigation is proposed as the Marine Electronic Highway (2013) to improve the navigation safety and security of maritime transportation. That consists of onboard and shore-based sensors and systems to communicate navigation information in the ship routes between Malacca and Singapore, which promotes sustainable development in marine and coastal environmental conservation among the states of Indonesia, Malaysia and Singapore. Furthermore, the e-navigation concept is further illustrated by various ongoing projects: “EfficienSea” (EfficienSea, 2013) and “MonaLisa” (MonaLisa, 2013). An important part of e-navigation consists of transforming ship navigation information into decision support features, therefore the information presentation and situation assessment features have also been considered in these projects. 2.2. Integrated bridge systems Integrated bridge systems (IBSs) can be considered as the main focus point of e-navigation, where the respective information sources (i.e. global and onboard) in ship navigation should appropriately be integrated. However, the proposed measures in e-navigation can eventually be evaluated under various complex ship maneuvers under collision and near collision situations with respect to the navigators’ decisions. Therefore, the best onboard navigation tools should be available to influence the navigators’ awareness of these high risk situations to increases the safety of maritime transportation, where the further improvements in present IBSs should be introduced. Hence, an overview of the present IBS technology is presented in this section. In general, IBSs consist of two separate networks: navigation and automation systems (see Fig. 1) that satisfy the Classification Society’s requirements and these separate networks

can improve the system survivability in various failure situations. The navigation system may consist of Radar, Conning and ECDIS (Electronic Chart Display and Information System), Auto-pilot system and other related sensors. The automation system may consist of power management architecture for engine and propulsion control systems that are related to various engine room operations. Additional units of bilge and ballast control, HVAC and alarm & monitoring systems can also be a part of the automation system. Ship Radar and ECDIS systems have extensively been used for detecting the collision risk (i.e. collision and near collision situations) among vessels in IBSs. Conventional Radar systems are facilitated by the ARPA (automatic radar plotting aid) capability that provides accurate information of the range and bearing values of nearby vessels in an electronic chart display with additional target tracking facilities. Modern Conning displays consist of ship position, course, speed, rudder, propeller, thruster, eco-sounder, wind and route information and the status of navigation lights to improve the safety of maritime transportation. One should note that ECDISs replace conventional paper maps by electronic charts and these charts consist of sophisticated digital displays that produce comprehensive environment information (i.e. shore information, ship information and hydrographic information) for the navigators. Those electronic charts are designed to meet the navigation requirements that are defined by the International Maritime Organization (IMO), International Electrotechnical Commission (IEC) and International Hydrographic Organization (IHO). Radar and ECDIS systems are facilitated with the Automatic Identification System (AIS) and the Long-Range Identification and Tracking (LRIT) system and both systems are capable of transmitting vessel identification and navigation information (i.e. position, heading, rate of turn, etc.) among vessels and shore based maritime authorities (IMO, 2003). Furthermore, these AIS and LRIT information has often been shared by vessel traffic monitoring and information systems (VTMISs) and that can also be used to detect potential collision situations among vessels. Therefore, IBSs should also collaborate with VTMISs and traffic separation schemes (TSSs) (Mou et al., 2010) to improve the navigation safety in maritime transportation (Perera et al., 2012b; Kao et al., 2007) and that can be another important step in e-navigation. TSSs are established in various

Fig. 1. Integrated bridge system.

L.P. Perera, C. Guedes Soares / Ocean Engineering 109 (2015) 344–354

congested water ways to reduce a considerable amount of collision/near collision situations. VTMISs are proposed as effective local vessel monitoring mechanisms for highly density maritime traffic regions, where vessels are informed for vessel traffic, collision risk, navigational difficulties and meteorological & environmental hazard, to manage ship routes within ports or waterways. However, these external safety systems (i.e. VTMISs and TSSs) are highly regulated by various local and international maritime rules and regulations. Even though, these navigation aids (i.e. GPS, AIS, LRIT, VTMIS and TSS) partially assist vessels to evaluate the collision risk in close encounter situations, appropriate collision detection facilities have not extensively been developed under IBSs. Hence, this requirement is highlighted under the e-navigation framework and illustrated as filtering the collision risk in maritime transportation. It is believed the collision risk detection and quantification (i.e. filtering the collision risk) methods can play an important role in the implementation phase of e-navigation and that will further enhance the navigation safety in maritime transportation (IMO, 2007). Therefore, those methods can be used to assess the collision risk among vessels ahead of close encounter situations and that information can be presented in IBSs as automated solutions to potential ship collisions (i.e. NAVDEC systems) by improving the situation awareness. Similarly, this study proposes to assess the collision risk in close encounter situations among vessels and that also supports the global vision of e-navigation with automated decisions support facilities with appropriate information presentations and situation assessments under IBSs.

3. Collision risk 3.1. Recent studies An important part of the e-navigation strategy consists of reducing ship navigation errors and that may also reduce the respective collision risk among vessels (Hahn, 2014), which eventually prevents shipping accidents and respective marine pollutions. Therefore, appropriate mathematical concepts should be developed to detect the respective risk among vessels in close encounter situations. The mathematical models for detecting the collision risk among vessels are divided into two main categories in the recent literature (Imazu, 2006): closest point approach (CPA) and predicted area of danger (PAD) approach. The CPA consists of calculating the shortest distance between two vessels and assessing the collision risk of one vessel (i.e. vessel domain) with respect to the other vessel route (Kwik, 1989). However, this collision risk assessment approach can degrade in some ship navigation situations, since it cannot consider the size, course and speed variations between vessels. The PAD approach consists of modeling the possible trajectories of one vessel as an inverted cone and other vessel possible trajectories as an inverted cylinder. The intersection region of both objects is categorized as the predicted area of danger for the respective vessels and the limited size, course and speed variations between vessels can be integrated into this approach. However, constant ship speed and course conditions are often assumed in both methods, therefore instantaneous variations in navigation conditions (i.e. navigation parameter variations) cannot be incorporated into these methods. Furthermore, these methods are simplified by assuming limited navigation conditions such as vessels moving in straight-line routes under deterministic state and parameter behavior (i.e. constant vessel speed and course conditions). Even though these assumptions can be applicable to other transportation systems, maritime transportation can often be involved with various not straight line maneuvers with under-actuated ship steering

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properties (Perera et al., 2015b) and stochastic state and parameter behavior due to varying sea conditions. Furthermore, various statistical analyses for detecting the collision risk among vessels are also presented in the recent literature. Ship collision situations are categorized by several probabilistic distributions with the statistical data from previous ship collisions (Rawson et al., 1998; Faghih-Roohi et al., 2014) in these approaches. However, the previous ship collision data may create an unreasonable base for future ship collision/near collision situations, because vessel structures, operational procedures, and navigation and communication technologies have significantly improved in the recent years (Pedersen, 2010). 3.2. Navigation challenges Various navigation challenges are faced by ship navigators during vessel close encounter situations, where the respective collision risk should be properly assessed. In general, ship navigators monitor collision/near collision situations by observing of the relative bearings of other vessels in the vicinity; the unchanged relative bearing of a vessel can often be resulted in a collision/near collision situations. However, this measurement (i.e. the relative bearing) can degrade the accuracy of the collision risk assessment between vessels in some navigation situations, because complex ship maneuvers can introduce additional challenges into the relative bearing calculation. i.e. when two vessels are under turning circle type maneuvers, the relative navigation trajectory of one vessel respect to the other vessel can be a complex ship route and that behavior may not be visible under this relative bearing calculation. Furthermore, the collision risk assessment can further be complicated by additional vessels that may also be in the vicinity. Therefore, this study proposes to overcome these navigation challenges by considering the following conditions under the proposed collision detection and quantification methodology, which can also be implemented under IBSs. First, AIS and GPS signals (i.e. vessel positions) are associated with known sensor noise and/or systems errors, therefore the position accuracy of the respective vessels can be further improved by an appropriate estimation algorithm. Second, the vessels are maneuvering under varying sea conditions, where the respective vessel states are associated with stochastic behavior (i.e. system noise). Third, the variations in course and speed values in the vessels are observed by a real-time monitoring process. One should note that these conditions are a part of the proposed collision detection and quantification methodology and that create realistic ship navigation conditions in close encounter situations. 3.3. Collision risk assessments A close encounter situation with two vessels, presented in Fig. 2, is considered in this section. A fixed XYZ right-hand coordinate system (i.e. North, East, Down reference frame) is derived in this situation. The proposed detection and quantification methodology consists of following steps:

 First, vessel positions (i.e. both vessels) are established.  Second, vessel velocity and acceleration components and  

navigational trajectories are estimated by an extended Kalman filter (EKF) with the position measurements. Third, the relative navigation trajectory (i.e. the relative coursespeed vector and the relative bearing vector of one vessel with respect to the other vessel) is estimated. Fourth, a vector dot and cross product based mechanism to assess the collision risk between vessels is introduced.

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motion model is proposed:

χ_ o ðtÞ ¼

ano ðtÞ V o ðtÞ

V_ o ðtÞ ¼ ato ðtÞ


 vxo ðtÞ ¼ V o ðtÞ cos χ o ðtÞ
 vyo ðtÞ ¼ V o ðtÞ sin χ o ðtÞ

ð1Þ

Hence, (1) can be summarized into the following system model: x_ x ðtÞ ¼ f ðxx ðtÞÞ þ wx ðtÞ h i E½wx ðtÞ ¼ 0; E wx ðtÞ; wx ðtÞT ¼ ½Q ðtÞ

ð2Þ

where xx(t) is the system state vector, and f(xx(t)) is the system function. That can also be summarized as 2 3 2 3 xo ðtÞ vxo ðtÞ 6 v ðtÞ 7 vxo vyo 7 6 6 xo 7 6 ato ðtÞf  ano ðtÞf 7 6 7 6 7 6 yo ðtÞ 7 6 vyo ðtÞ 7 6 7 6 7 xx ðtÞ ¼ 6 ; f ðxx ðtÞÞ ¼ 6 7 vyo vxo 7 6 vyo ðtÞ 7 ðtÞf þ a ðtÞf a to no 6 7 6 7 6 7 6 ato ðtÞ 7 40 5 4 5 ano ðtÞ 0 Fig. 2. Two vessel collision situation.

 Fifth, a transformation to observe the relative navigation 

trajectory of one vessel with respect to the other vessel position, course and heading conditions is presented. Finally, the above results are used to evaluate and visualize the collision risk between two vessels.

One should note that there are some similarities between the estimated relative navigation trajectory approach (i.e. the proposed approach) and the conventional bearing observation method for ship encounter situations. However, the relative course-speed vector of a vessel can be considered as an additional tool to evaluate the collision risk between these vessels. Even though this study is limited to a ship encounter situation with two vessels, this approach can be expanded to ship encounters with multiple vessels by accumulating several two vessel encounter situations as shown in Perera et al. (2012c).

4. Mathematical model As presented in Fig. 2, the close encounter situations consists of two vessels: vessel O is located in O (xo(t), yo(y)) and vessel A is located in Pa (xa(t), ya(t)). Vessels O and A course and speed values are χo(t) & Vo(t) and χa(t) & Va(t), respectively. The X and Y velocity vectors of vessels O and A are vxo(t), & vyo(t) and vxa(t) & vya(t), respectively. The normal and tangential acceleration vectors of vessels O and A are ano(t) & ato(t), and ana(t) & ata(t), respectively. The relative course-speed vector of vessel A with respect to vessel O is Vao(k). The estimated bearing vectors of vessels O and A are Bpo(k) and Bpa(k) (that are calculated in discreet-time instants). The heading vector of vessel O is Ho(k). The relative bearing vector of vessel A with respect to vessel O is Bao(k). The vessel O domain is Rvd. Furthermore, additional vectors that are in this figure are described in the respective sections. 4.1. System model A mathematical model that can capture realistic vessel maneuvering behavior is considered in this section. To capture the maneuvering trajectory of vessel O, a continuous-time curvilinear

f

vxo

vxo ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 vxo ðtÞ þ v2yo ðtÞ

f

vyo

vyo ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vxo ðtÞ þ v2yo ðtÞ

ð3Þ

and wx(t) is white Gaussian process noise with 0 mean and Q(t) variance. The Jacobian of the system function, f(xx(k)), can be expressed as ∂ ∂xx ðf ðxx ðtÞÞÞ ¼

2

0 60 6 6 60 6 60 6 6 40

1 vxo vyo ato ðtÞf vxo ano ðtÞf vxo

0

0

0

0

0

vxo vyo ato ðtÞf vyo  ano ðtÞf vyo

f

vxo

f

vyo

0 f

0

1

ato ðtÞf vxo þ ano ðtÞf vxo

0

ato ðtÞf vyo þ ano ðtÞf vyo

0

0

0

0

0

0

0

0

0

0

vyo

0

vxo

vyo

0

vyo

vxo

0 f

vxo

3 7 7 7 7 7 7 7 7 5

ð4Þ where the respective functions in (4) can be written as v2yo ðtÞ

vxo

f vxo ¼  vyo

3=2 ;

v2xo ðtÞ þv2yo ðtÞ

f vxo ¼  

vxo ðtÞvyo ðtÞ 3=2 ; v2xo ðtÞ þ v2yo ðtÞ

vxo

f vyo ¼   vyo

vyo ðtÞvxo ðtÞ 3=2

v2xo ðtÞ þ v2yo ðtÞ

f vyo ¼ 

v2xo ðtÞ v2xo ðtÞ þ v2yo ðtÞ

3=2

ð5Þ

4.2. Measurement model A mathematical model that observes the measured vessel positions is considered. The respective vessel position measurements can be extracted from AIS, GPS and Radar information as discussed before (Perera et al., 2012d; Ferrari et al., 2015). A discrete-time measurement model is considered in this section due to the availability of vessel position values in discrete time instants. Hence, the measurement model of vessel O can be written as zz ðkÞ ¼ hðxx ðkÞÞ þ wz ðkÞ; k ¼ 1; 2; … h i     E wz ðkÞ ¼ 0; E wz ðkÞ; wz ðkÞT ¼ RðkÞ

ð6Þ

where zz(k) is the measurement state vector, and h(xx(k)) is the measurement function. That can be further described as " # " # zx ðkÞ 0 0 0 0 xo ðkÞ 0 ; hðxx ðkÞÞ ¼ zz ðkÞ ¼ ð7Þ zy ðkÞ 0 0 yo ðkÞ 0 0 0

L.P. Perera, C. Guedes Soares / Ocean Engineering 109 (2015) 344–354

where zx(k) and zy(k) are the X and Y position measurement values of vessel O, and wz(k) is white Gaussian measurement noise with 0 mean and R(k) variance. The Jacobian matrix of the measurement function, h(xx(k)), can be written as

1 0 0 0 0 0 ∂ ðhðxx ðkÞÞÞ ¼ ð8Þ ∂xx 0 0 1 0 0 0 A negligible correlation between the system and measurement noises is assumed and that can be written as h i ð9Þ E wx ðtÞ; wz ðkÞT ¼ 0 for all k; t Furthermore, similar systems and measurement models are developed for vessel A to estimate the respective vessel states. 4.3. Extended Kalman filter (EKF) The system states (i.e. the position, velocity and acceleration components) for both vessels from the position measurements are estimated by considering an EKF algorithm. The estimated vessel states are used to derive the relative navigation trajectories, the relative course-speed vectors and the relative bearing vectors of one vessel with respect to the other vessel and that are used to evaluate the respective collision risk between the vessels. Even though the vessel position measurements consist of the respective sensor noise, the estimation algorithm (i.e. EKF) can accommodate that information (i.e. sensor noise) to improve the vessel position accuracy. A overview of an EKF algorithm is presented in the following section and the position error, x~ x ðkÞ, in the vessel states can be written as (Gelb et al., 2001) x~ x ðkÞ ¼ x^ x ðkÞ  xx ðkÞ

ð10Þ

where x^ x ðtÞ is the estimated system state vector. Therefore, the respective initial state conditions can be written as
 ð11Þ xx ð0Þ  N x^ x ð0Þ; Pð0Þ where P(0) is the initial state covariance. Then, the state estimate propagation can be written as: d x^ x ðkÞ ¼ f ðx^ x ðkÞÞ dt

ð12Þ

where the function, Fðx^ x ðkÞÞ, can be written as   ∂ f ðxx ðkÞÞ Fðx^ x ðkÞÞ ¼ ∂xx ðkÞ xx ðkÞ ¼ x^ x ðkÞ

ð13Þ

ð14Þ

and P(k) is the estimated error covariance with the estimated state updates. The vessel states are updated by the respective measurement data and that can be written as 
þ    x^ x ðk Þ ¼ x^ x ðk Þ þ KðkÞ zðkÞ  hk x^ x ðk Þ ð15Þ þ

-

where xx(k ) and xx(k ) are the prior and posterior estimated system states, and K(k) is the Kalman gain, respectively. The error covariance updates can be written as 
þ    Pðk Þ ¼ 1  KðkÞH x^ x ðk Þ Pðk Þ ð16Þ -

þ

where P(k ) and P(k ) are the prior and posterior error covariances of the system estates, respectively and the function, Hðx^ x ðkÞÞ, can be written as   ∂ hðxx ðkÞÞ Hðx^ x ðkÞÞ ¼ ð17Þ ∂xx ðkÞ xx ðkÞ ¼ x^ x ðkÞ

ð18Þ

4.4. Collision risk assessment The collision risk between the vessels is assessed in this section. That is done by deriving a relationship between the bearing and relative course-speed vectors of one vessel with respect to the other vessel position in this vessel encounter situation. Therefore, the relative position vector components of vessel A with respect to vessel O can be written as x^ ao ðkÞ ¼ x^ a ðkÞ  x^ o ðkÞ y^ ao ðkÞ ¼ y^ a ðkÞ  y^ o ðkÞ

ð19Þ

where x^ ao ðkÞ and y^ ao ðkÞ are the estimated X and Y relative coordinates of vessel A with respect to vessel O and the respective position vectors of vessels A and O are denoted by Bpa ðkÞ and Bpo ðkÞ. One should note that (19) represents the relative navigation trajectory of vessel A with respect to vessel O. The relative course-speed vector components of vessel A with respect to vessel O can be written as v^ xao ðkÞ ¼ v^ xa ðkÞ  v^ xo ðkÞ v^ yao ðkÞ ¼ v^ ya ðkÞ  v^ yo ðkÞ

ð20Þ

where v^ xao ðkÞ and v^ yao ðkÞ are the estimated X and Y relative velocity components (i.e. the relative course-speed vector) of vessel A with respect to vessel O. Hence, the estimated relative course-speed vector of vessel A with respect to vessel O can be presented as h i V ao ðkÞ ¼ v^ xao ðkÞ v^ yao ðkÞ 0 ð21Þ The unit vector of (21) can be written as V ao ðkÞ ¼ V ao ðkÞ=‖V ao ðkÞ‖2

ð22Þ

The estimated relative bearing vector of vessel A with respect to vessel O can be written as h i ð23Þ Bao ðkÞ ¼ x^ ao ðkÞ y^ ao ðkÞ 0 Similarly, the unit vector of (23) can be written as Bao ðkÞ ¼ Bao ðkÞ=‖Bao ðkÞ‖2

The error covariance extrapolation can be written as d PðkÞ ¼ Fðx^ x ðkÞÞPðkÞ þ PðkÞF T ðx^ x ðkÞÞ þ Q ðkÞ dt

The Kalman gain, K(k), can be calculated by h
i1

       T KðkÞ ¼ Pðk ÞH x^ x ðk Þ H x^ x ðk Þ Pðk ÞH x^ x ðk Þ þ RðkÞ

349

ð24Þ

One should note that Gao ðkÞ (i.e. the unit vector of Gao ðkÞ) is perpendicular to Bao ðkÞ and that relationship can be also denoted by the following vector cross product: Go ¼ ZðkÞ  Bao ðkÞ

ð25Þ

where ZðkÞ is the unit vector of Z axis and that can be written as   ZðkÞ ¼ 0 0 1 ð26Þ A vector with the vessel domain magnitude and Go ðkÞ direction can be written as Go ðkÞ ¼ Rvd Go ðkÞ

ð27Þ

Therefore, the boundaries of the vessel O domain with respect to vessel A (see Fig. 2) can be denoted by two vector of F oa ðkÞ and C oa ðkÞ (see Fig. 2) and that can be written as F oa ðkÞ ¼ Go ðkÞ  Bao ðkÞ C oa ðkÞ ¼  Go ðkÞ  Bao ðkÞ

ð28Þ

where the respective unit magnitude vectors of (28) can be written as F oa ðkÞ ¼ F oa ðkÞ=‖F oa ðkÞ‖2 C oa ðkÞ ¼ C oa ðkÞ=‖C oa ðkÞ‖2

ð29Þ

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Hence, the collision risk between two vessels can be calculated by considering V ao ðkÞ, which should locate between F oa ðkÞ and C oa ðkÞ to create a collision situation. This condition can be mathematically denoted as a vector product of
 F oa ðkÞ  V ao ðkÞ U ZðkÞ Z0  
 C oa ðkÞ  V ao ðkÞ U  ZðkÞ Z 0 ð30Þ Therefore, the collision risk in (30) is assessed by deriving a relationship between the bearing vector and the relative coursespeed of one vessel with respect to the other vessel in this close encounter situation. The vector product between the unit vectors of the relative course-speed and bearing vectors (i.e. of vessel A) is categorized as the collision risk (CR) between the vessels and that can be written as
 CRðkÞ ¼ V ao ðkÞ U Bao ðkÞ Z 0 ð31Þ Therefore, the conditions presented in (30) and (31) should satisfy to identify a potential collision/near collision situation between the vessels and that are further illustrated in the following computational simulation. If a two vessel encounter situation satisfies the condition in (30), then the respective collision risk in (31) should be assessed. The highest and lowest collision risk values in (31) can be denoted by 1 (i.e. the relative course-speed vector and the relative bearing vector are parallel with opposite directions) and 0 (i.e. the relative course-speed and the relative bearing vectors are perpendicular), respectively. 4.5. Trajectory transformation Finally, the relative navigation trajectory of one vessel with respect to the other vessel heading is estimated. That can be done by calculating the relative position of vessel A with respect to vessel O heading. A vector perpendicular to vessel O heading, K o ðkÞ, can be written as K o ðkÞ ¼ H o ðkÞ  ZðkÞ

ð32Þ

where H o ðkÞ is the heading vector of vessel O and the respective unit vector can be written as h i ð33Þ H o ðkÞ ¼ H o ðkÞ =‖H o ‖ðkÞ2 ¼ hox ðkÞ hoy ðkÞ 0 where hox ðkÞ and hoy ðkÞ are the respective X and Y vector components. The unit vector of (32) can be written as h i ð34Þ K o ðkÞ ¼ K o ðkÞ=‖K o ðkÞ‖2 ¼ kox ðkÞ koy ðkÞ 0 where kox ðkÞ and koy ðkÞ are the respective X and Y vector components. The heading vector of vessel O can be transformed to the X axis by considering the vector summation of

αo ðkÞXðkÞ ¼ Ho ðkÞ þ βo ðkÞK o ðkÞ

ð35Þ

where αo ðkÞ and β o ðkÞ are the respective transformation parameters. The unit vector in X axis can be written as   ð36Þ XðkÞ ¼ 1 0 0 The heading vector of vessel O can be defined as h i H o ðkÞ ¼ Rvd H o ðkÞ ¼ hox ðkÞ hoy ðkÞ 0

βo ðkÞ ¼  hoy ðkÞ=koy ðkÞ

A unit vector perpendicular to the vessel A relative bearing can be calculated as J o ðkÞ ¼ Bao ðkÞ  ZðkÞ

Bo ðkÞ ¼ Rvd Bao ðkÞ þ βo ðkÞJ o ðkÞ

ð41Þ

where, the respective unit vector of (41) can be written as Bo ðkÞ ¼ Bo ðkÞ=‖Bo ðkÞ‖2

ð42Þ

Hence, the transformed relative position vector of vessel A with respect to the vessel O heading can be written as (see Fig. 2) P o  sgnðαo Þ‖Bao ðkÞ‖2 Bo ðkÞ

ð43Þ

5. Computational simulations A two vessel encounter situation is presented in Fig. 3. The figure consists of the actual (Act.), measured (Mea.) and estimated (Est.) vessel positions (i.e. navigation trajectories (Traj.)) of vessels O and A, respectively. A zoomed view of the same navigation trajectories around the closest encounter position of the vessels is presented in Fig. 4. The actual vessel position values are simulated by assuming the derivatives of vessel acceleration components as white Gaussian processes with zero mean and constant covariance values (Perera et al., 2012b). The measured vessel position values are generated by adding sensor noise into the actual vessel position values and an EKF algorithm is used to calculate the estimated vessel position values. One should note that various nonlinear trajectories can be created by this approach and that can be used to approximate complex vessel maneuvering conditions (i.e. that is also independent from the shapes of vessels trajectories). An EKF algorithm with a continuous-time curvilinear motion is proposed in this study and that can capture actual vessel maneuvering behavior with stochastic state conditions. Furthermore, the EKF algorithm is highly effective in the situations, where the system-model uncertainties are negligible. However, this approach may have some challenges under other navigation systems (i.e. aircrafts), where highly discrete or nonlinear maneuvers can be observed. Furthermore, the EKF algorithm needs less computational power with comparison to other nonlinear estimation algorithms, which can also contribute to improve the convergence time of the algorithm. However, the convergence time for the EKF algorithm may vary in different voyage segments due to their shapes and the initial state conditions.

ð37Þ

ð38Þ

Hence, (38) can also be simplified as

αo ðkÞ ¼ hox ðkÞ  hoy ðkÞkox ðkÞ=koy ðkÞ

ð40Þ

Hence, the relative position vector of vessel A with respect to the vessel O heading can also be transformed along Bo ðkÞ and that can be written as

The parameters, αo ðkÞ and βo ðkÞ, are derived by considering (34)–(37), and the respective vector components can be written as

αo ðkÞ ¼ hox ðkÞ þ βo ðkÞkox ðkÞ 0 ¼ hoy ðkÞ þ β o ðkÞkoy ðkÞ

ð39Þ

Fig. 3. Navigation trajectories vessel A and vessel O.

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Fig. 4. A zoomed view navigation trajectories vessel A and vessel O.

Fig. 6. Collision risk.

Fig. 5. The relative trajectory of vessel A w/r vessel O position.

The estimated relative trajectory (Est. Rel. Traj.) and the respective relative course speed vectors (i.e. that presented by arrows) of vessel A with respect (w/r) to vessel O are presented in Fig. 5. This is an important representation of the vessel encounter situation and that (i.e. the relative course-speed vector) shows either vessels move towards or away from each other. Therefore, that information can be extremely useful in a close encounter situation and that can also be used to assess the respective collision risk between the vessels. As presented in the figure, the vessel O domain is denoted as a circular region with diameter of 400 m and vessel O is located in the center of this domain. One should note that the vessel domain can be a vessel related value, which may vary from the respective ship length to 2 nautical miles under various navigation conditions (i.e. coastal areas to open sea regions). Furthermore, the respective course-speed vectors of vessel O are also presented in the same figure (i.e. presented by arrows) at the vessel initial position. However, the relative coursespeed vectors in Fig. 5 are scaled appropriately to improve the visibility of the vessel encounter situation. One should note that an intersection of two trajectories cannot be concluded as being a collision point because each vessel can pass this point in different time intervals. However, this confusion can be clarified by observing the relative navigation trajectories of one vessel with respect to the other vessel in a close encounter. An intersection between the relative navigation trajectory of vessel A

and the vessel O domain is considered as a collision situation and a close encounter of the relative navigation trajectory of vessel A with the vessel O domain is considered as a near collision situation in this study. Therefore, the situation in Fig. 5 is categorized as a ship collision situation and another collision situation is presented in Perera and Guedes Soares (2012), which consists of additional computational simulations to support the proposed methodology. Furthermore, the calculated collision risk in (31) with respect to the relative distance between the vessels is considered in Fig. 6. The respective collision risk is presented as a distance (i.e. between the vessels) based plot to improve the visibility of the ship encounter situation in this study. However, this is a novel approach and that is slightly deviated from the conventional collision risk models (i.e. probability multiplies consequence). Hence, the respective increments and decrements in the collision risk are monitored with respect to the distance between the vessels in this approach. That (i.e. the collision risk) should also be compared with the relative navigation trajectory and the relative course speed vector of vessel A with respect to vessel O (see Fig. 5) to understand the severity of the vessel encounter situation. One can also conclude that the collision risk between these two vessels can be observed long before the vessel encounter situation by observing Figs. 5 and 6, where the appropriate collision avoidance actions can be executed. The estimated collision risk (see Fig. 6) should be compared with appropriate severity levels, and that should be categorized with respect to the vessel O domain. Therefore, the navigators can decide various collision avoidance actions with respect to these risk levels (i.e. severity), where the immediate collision avoidance actions should be taken for the highest risk level. The encounter distance between the vessels can also play an important role in this approach, where far-away vessels with higher collision risk levels can be ignored. Hence, the relative distance based collision risk assessment (see Fig. 6) can be categorized as an important contribution in this study and that can also be an important part of the future IBSs. There are several collision risk peaks can be observed around (yao ¼)  1000 m and 0 m (see Fig. 6). The collision risk peak around yao ¼ 1000 m represents the initial position of vessel A, where its relative course-speed vector has changed several times (see Fig. 5). However, the collision risk peak around yao ¼ 0 m represents the actual collision situation. These collision risk peaks can be ignored, if the relative distance of the vessels is significantly larger with compare to the respective vessel domain. Therefore,

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the collision risk peak around yao ¼  1000 m is ignored and yao ¼0 m is considered as a ship encounter situation with the highest collision risk. One should note that this method helps ship navigators to make proper collision avoidance decisions ahead of close vessel encounter situations, where the collision risk variations due to their actions can also be observed. Hence, Figs. 5 and 6 should compare simultaneously to see a complete picture of the vessel encounter situation, where the collision risk between the vessels with respect to the relative distance can be varied. This relative distance based collision risk approach may slightly deviate from other conventional time based collision risk evaluations methods. The time period ahead of a vessel close encounter may not be a constant in some situations, where the respective course-speed vectors of the vessels can continuously vary. The respective time period ahead of a possible collision situation should be continuously calculated to update the risk severity level by considering the relative navigation trajectory and the respective course speed vector of the vessels in these situations. Hence, it is believed that this approach (i.e. the collision risk with respect to the relative distance) can improve the visibility of the collision risk in a close encounter situation under several severity levels. The proposed distance based collision risk assessment approach also shows consistence collision risk variations in complex ship encounters (i.e. even with varying course-speed vectors) and that is another advantage, where conventional collision risk assessment methods may fail to identify. The estimated relative navigation trajectory (Est. Rel. Traj.) of vessel A with respect to the vessel O heading for the same situation is presented Fig. 7. It is assumed that the vessel O heading is varying from 0 rad) to π =2 rad) in a constant turning rate in this situation and those heading variations (i.e. the arrows) are also presented in the vessel O position in the same figure. However, these heading vectors are also appropriately scaled to improve the visibility of the collision situation. The same vessel encounter situation is further elaborated in Fig. 7, where each estimated relative position of vessel A with respect to the vessel O heading is presented by a red color dot, therefore vessel A intercepts the vessel O domain from starboard. One should note that the respective collision avoidance actions accordance with the international regulations for preventing collisions at sea (COLREGs) should be taken by the navigator with respect to the approaching direction of other vessels. This approach influences the decision making process of the navigator with the most important information on a ship encounter situation, where the approaching vessel (that is intercepting the vessel domain)

Fig. 8. Vessel velocities.

direction (i.e. either starboard or port) can be determined. Therefore, the appropriate COLREGs rules and regulations can be implemented by the navigator with respect to the approaching vessel direction. The respective actual (Act.) and estimated (Est.) velocity components of vessels O and A are presented in Fig. 8. The respective actual and estimated acceleration components of the same vessel are presented in Fig. 9. Even though an EKF algorithm has diverged at some initial estimation steps, which has successfully converged to the actual velocity and acceleration components of both vessels at the following estimation steps. However, some slight variations on the convergence times of the EKF algorithm can be observed in this simulation due to the shape of vessel trajectories and the initial vessel states as mentioned previously. The EKF algorithm can also used under discrete time intervals in these situations, where the estimated states of one time interval can be used as the initial state values for the next time interval. Therefore, that process can reduce the state estimation time for the EKF algorithm.

6. Conclusions

Fig. 7. The relative trajectory of vessel A w/r vessel O heading.

A methodology for assessing the collision risk in an enavigation environment with vessel state uncertainties (i.e. stochastic behavior) under complex maneuvers is presented in this study. The proposed approach is evaluated by estimating the relative navigation trajectory, the relative course-speed vector and the relative bearing vector of one vessel with respect to the other vessel in a two vessel encounter situation. The proposed methodology can improve the awareness of the situation in the bridge (i.e. IBSs) due to this vessel relative distance based risk detection and quantification methodology.

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on assessing the collision risk in a ship encounter situation are reported. However, various ship navigation vectors (i.e. vessel states) are introduced in this study that can somewhat relate to the navigation characteristics of vessels as well as onboard sensor capabilities (i.e., radar and laser systems) (Perera et al., 2014). Therefore, the parameters in those navigation vectors should be further investigated and that is also proposed as a part of future studies. This study is a continuation of the collision avoidance studies that are presented in Perera et al. (2015a, 2012c), expanding earlier work to assess the collision risk in an e-navigation environment with vessel state uncertainties (i.e. stochastic behavior) under complex maneuvers. The proposed approach plays an important role in the future of IBSs to improve the navigation safety in maritime transportation and that can further enhance the enavigation functionalities. However, further experiments should be conducted to further evaluate the proposed vessel relative distance based collision risk detection and quantification methodology and that are also proposed as the future work of this study.

Acknowledgement

Fig. 9. Vessel accelerations.

The proposed collision risk detection and quantification methodology is simulated and evaluated under a two vessel encounter (i.e. collision/near-collision) situation and the respective results are also discussed in this study. One should note that constant vessel state conditions are assumed by a majority of other conventional collision risk detection methods, where the instantaneous changes on course and speed conditions among vessels cannot be incorporated. This study is not based on that assumption, therefore it can be adopted for ship encounter situations with vessel state variations under complex ship maneuvers (i.e. realistic ship navigation conditions). The proposed methodology may be less effective in ship close encounter situations especially in restricted navigation areas (i.e. rivers and channels), where the navigator’s decisions can be limited and the vessel controllability may also be inadequate. However, the relative course-speed vector information can still be used in these situations to observe whether vessels moving towards or away from each other and that information can be an important tool in the navigation decision making process. An EKF algorithm with a continuous-time curvilinear motion model is used in this study to estimate the vessel position, velocity and acceleration components of the vessels. Even though these ship acceleration components have not been extensively used in current ship navigation systems (i.e. IBSs), this study shows that several accelerometers can be placed in vessels and that information (i.e. vessel acceleration in several hull positions) can be used to improve the navigation safety. e.g. the estimated vessel position and velocity components can be improved. Therefore, these improved vessel states can be used to define appropriate severity levels for the respective collision risk (see Fig. 6), where the navigator’s awareness of the close ship encounter situations can be further improved. The proposed approach capabilities are evaluated under the respective computational simulations, where the successful results

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