# Estimating Critical Latency Affecting Ship's Collision in Re-Mote Maneuvering of Autonomous Ships
## 2 Definition of Concepts and Terms Regarding Critical Latency
### Concepts for Critical Latency Estimation  
The concepts for critical latency estimation are described in Figure 2. The concept in Figure 1 that considers the crossing situation is based on the collision relationship between the OS and other ships, which may appear if the rudder control command given to the OS is delayed. In addition, in this concept, the ship's scale (i.e., length, width) is assumed to be zero to explain the concept of critical latency.  
Figure 2: Concept for estimating a critical latency.  
Many small circles represent the positional relationship over time between the OS and other ships. The four small circles shown in O1-O3 represent the points at which the OS can be located while traveling the distance O1-O3 along the given path (long solid line in the longitudinal direction), and the five small circles shown in T0-T3 represent the points at which the other ship can be located while traveling the distance T0-T3 along the given path (long solid line in the lateral direction). In addition, the distances among the three circles--O1, O2, and O3--and the distances among the four circles--T0, T1, T2, and T3--are assumed to be equidistant. The three points--C, M1, and M2--represent the possible collision points between the OS and other ships. C represents the point of collision caused by the encountering situation where O3 and T3 overlap. M1, represented as a red circle, is the point where the OS turning maneuvered at O2 can collide with the other ship that is at T2, and M2, represented as a red circle, is the point where the OS turning maneuvered at O23 can collide with the other ship, that is, at T23. \(\Gamma\)1 and \(\Gamma\)2 represent the latency of O1-O2 and O1-O23, respectively, \(\left\{\mathbb{t}\,\mathbb{t}\,\mathbb{1}\,\mathbb{2}\right\}\)_turn TS and \(\left\{\mathbb{t}\,\mathbb{2}\right\}\)_turn TS represent the travel time of T0-T2 and T1-T23, respectively, and \(\left\{\mathbb{t}\,\mathbb{1}\,\mathbb{2}\right\}\)_turn and \(\left\{\mathbb{t}\,\mathbb{2}\right\}\)_turn represent the turning time of O2-T2 and O23-T23, respectively. The two collision Points--M1 and M2--can occur in two postulated situations--Situations 1 and 2:  
Situation 1 is when the OS travels along the path of O1-O2-T2, while the other ship travels along the path of T0-T1-T2, causing a collision at M1. This situation may occur if the rudder control command given to the OS at O1 to avoid collision is operated at O2 due to \(\Gamma\)1.  
Situation 2 is when the OS travels and turns along the path of O1-O23-T23, while the other ship travels along the path of T1-T2-T23, causing a collision at M2. This situation can occur if the rudder control command given to the OS at O1 to avoid collision is operated at O23 due to \(\Gamma\)2.  
Point M1 or M2 can be varied according to various circumstances. Point M1 can occur only when the positions of the two ships are in contact with the turning circle, and Point M2 can occur in a variety of ways at M1-C. In addition, of the two latencies, \(\Gamma\)1 becomes minimal and \(\Gamma\)2 becomes maximum, and if collision avoidance is to be considered, the latency is required to be minimal (zero in theory). The critical latency defined in this study refers to the minimum latency among the latencies due to which two vessels may collide. The characteristic of this critical latency is that it can occur only when the positions of the two vessels are in contact with the turning circle. In this study, the critical latency is estimated.