The understanding and interpretation of simplicty in the context of a navigational path can be ambiguous. Different approaches to path planning are briefly juxtaposed, focusing on their simplicity-based distinctive features. This takes into consideration the response to drift caused by air or water currents and the geometric background. Special attention is paid to the straight-line-based solutions which are preferred by the human convenience and perception, recalling some well-known applications in navigation and including their (dis)advantages. In particular, this refers to the Euclidean simplicity applied at the cost of effectiveness in the standard search and rescue (SAR) patterns in the presence of relatively stronger position-dependent currents.

This paper attempts to solve a challenge in online relative optimal path planning of unmanned surface vehicles (USVs) caused by current and wave disturbance in the practical marine environment. The asymptotically optimal rapidly extending random tree (RRT*) method for local path optimisation is improved. Based on that, an online path planning (OPP) scheme is proposed according to the USV's kinematic and dynamic model. The execution efficiency of RRT* is improved by reduction of the sampling space that is used for randomly learning environmental knowledge. A heuristic sampling scheme is proposed based on the proportional navigation guidance (PNG) method that is used to enable the OPP procedure to utilise the reference information of the global path. Meanwhile, PNG is used to guide RRT* in generating feasible paths with a small amount of gentle turns. The dynamic obstacle avoidance problem is also investigated based on the International Regulations for Preventing Collisions at Sea. Case studies demonstrate that the proposed method efficiently plans paths that are relatively easier to execute and lower in fuel expenditure than traditional schemes. The dynamic obstacle avoidance ability of the proposed scheme is also attested.

Procedures based on solutions of the direct and inverse geodetic problems are presented and their application in orthodromic and loxodromic navigation is analysed. They can be used in traditional navigation during a route planning and during sailing on the planned route in real time. The optimal division of orthodromes during planning and locally predicted short loxodromes during sailing – which allow orthodromic navigation also in manual steering – are proposed.