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APPENDIX
Consider the movement of two vessels, own ship and an external target i, traveling on a plan sea surface in an Earth-fixed Cartesian co-ordinate system x-y. It is assumed that the vessels are mass points without extension and that the velocity vectors can be regarded as constant relative to the Earth-fixed co-ordinate. The movement vectors are furthermore considered as known at any time. The two vessel's encounter geometry is plotted in Fig. 3.
Let Xo(L) = [Xo Yo]T and XTi(L) =[XTi YTi]T be the respective position vectors of the vessels, Vo(LT-1) = [Vox Voy]T and VTi(LT-1) = [VTxi YTyi]T represent the corresponding true velocity vectors, and let ψO and ψT represent the respective ture courses. Furthermore, t (T) is time, Δt (T) is equivalent with the vector length in time scale, and α is the aspect angle (relative bearing). The relative vector VRi(LT-1) is then, on component form, given by
<ViRx,ViRy> = <ViTx,ViTy> - <VOx,VOy> (3)
The time to, and minimum distance at, closest point of approach, referred to as TCPA (T) and DCPA (L), respectively, follows from minimizing the time derivative of the relative distance function,
and can be written as
and
The cone-shaped collision danger region (Fig. 4) in true motion display is determined by the position vectors Xki(L) = {[Xki Yki]T}k = {A,B,C,D} which can be formulated as,
XiA = XiT + ViTΔtIΨi (7)
and
XiB = XO + ViTΔtIΨi (8)
where Ψi = [sinΨTicosΨTi]T for O ≤ ΨTi ≤ 2π and I is the 2x2 unit matrix,
and
where
Ωi =[(-1)mcos(γi + (-1)nβi(-1)ρsin(γi + (-1)nβi)]T
Φi =[(-1)mcos(γi + (-1)qβi(-1)ρsin(γi + (-1)qβi)]T
m, n, ρ and q are integers (0 or 1 ) according to;
YAi ≥ YBi and XAi ≥ XBi : [m, n, p, q] = [0, 1, 0, 0]
YAi ≥ YBi and XAi < XBi : [m, n, p, q] = [1, 1, 0, 0]
YAi < YBi and XAi ≥ XBi : [m, n, p, q] = [0, 0, 1, 1]
YAi < YBi and XAi < XBi : [m, n, p, q] = [1, 0, 1, 1]
AUTHOR'S BIOGRAPHY
Egil Pedersen received a doctorate in Nautical Science from the Department of Marine Technology at the Norwegian University of Science and Technology (NTNU) in 1997. He is a fully trained merchant ship navigational officer. Dr. Pedersen is currently a visiting researcher to National Maritime Research Institute (NMRI) in Tokyo.
Junji Fukutois Head of Navigation System Group at NMRI. He got a doctorate from Department of Naval Architecture, Ocean Engineering and Engineering Systems, Graduate school of Hiroshima University in 2001. Dr. Fukuto has been engaged in research on development of full-automated ship, safety assessment of HSC navigation and development of navigation support system. He is currently working with developing an automatic collision and grounding avoidance system.
Masayoshi Numano gained a Master of Engineering degree at the graduate school of Kyoto University. He has been engaged in research on energy saving, prevention of pollution, automation and human interface at NMRI since 1978.
Hiroko Itoh holds a position as researcher at NMRI. She majored in Engineering at the University of Tokyo where she received her Ph.D in 2001. Dr. Itoh has been involved in sea traffic simulation development, ship maneuverability assessment and human factors analysis.
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