# Estimating Critical Latency Affecting Ship's Collision in Re-Mote Maneuvering of Autonomous Ships
## 2 Definition of Concepts and Terms Regarding Critical Latency
### Coordinates and Terms  
The concept of estimating critical latency that can cause collisions has been established using turning circles of the own ship, which can be explained in the x-y plane coordinates. Figure 1 illustrates both the turning circle of the own ship and (OS-TS) positional relationship between the own ship (OS) and target ship (TS).  
A large circle represented by a solid-black line depicts a turning circle with a radius R from the coordinate center O (0,0). The encountering situation of OS and TS is assumed to be a crossing situation, where the two ships collide at Point C on given routes. The definition of each symbol shown in the coordinates is as follows:  
* \(\phi_{\text{OS}}\) and \(\phi_{\text{TS}}\), OS's heading (90\({}^{\circ}\)) and other ship's heading (180\({}^{\circ}\));
* O1 and O2, OS's arbitrary location (x, y) that appears over time;  
Figure 1: Description of relationship between own ship and target ship while introducing the turning circle of the own ship.  
* T1 and T2, TS's arbitrary location (x, y) that appears over time;
* RD, OS-TS relative distance (m);
* RB, OS-TS relative bearing (radian);
* \(\delta\), OS's rudder angle (\({}^{\circ}\));
* C, point where the two ships may collide under their encountering situation;
* S, point where the OS starts turning;
* M, point where OS, which has turned at Point S, may collide with TS;
* TC, turning circle with a radius R (m) from the coordinate center O;
* \(t_{\text{turn}}\), turning time (second or minute) of S-M;
* \(\theta_{\text{TC}}\), central angle (radian) of \(t_{\text{turn}}\) at O;
* \(\Gamma\), critical latency (second or minute).  
In Figure 1, in the positional relationship between the own ship and target ship, in accordance with COLREGs Rule 15 (crossing situations), the OS becomes a give-way vessel and the TS becomes a stand-on vessel. Where COLREGs are applied, the OS shall do a turning maneuver to its starboard side to avoid a collision. Generally, on ships, a turning maneuver is performed using the rudder. The turning maneuver results appear above a turning circle with radius R (m), whose scale is determined by the rudder angle applied to the turning. The long, red arc represents the S-M section formed by the turning maneuver, and the OS and TS collide at Point M.  
One of the possible causes for this Point M is the circumstance where due to a certain latency, the OS cannot be turning maneuvered at O1 but is turning maneuvered at O2. Furthermore, it is known that such a latency can be caused by various environments (i.e., communication network, control signal processing, human factors) in the case of remote maneuvering where communication networks are utilized [4; 5; 6; 7; 8; 9]. In addition, the positions where the M1 or M2 points can occur may be varied by the degree of latency and/or by various environments (i.e., the encountering situation, rudder angle, speed, relative distance between the two vessels) [16; 17; 18; 19; 20; 21]. The concept of estimating critical latency resulting from these various environmental changes will be discussed in Section 2.3.