# A New Understanding and Modelling of TSP and BP Indices Compared to Safety IMO Ship Requirements
## 1 Introduction  
For several decades, towing operations have been lucrative commercial activities and a relevant key of safety in the shipping industry [1, 2]. Furthermore, the growing significance of the size of ships in the last years, which does not match with the increase of working areas of ports, makes it evident that the margin to safety has been reduced [3]. Therefore, considering that nowadays towing operations (ship assistance and escort towing) are vital from a safety point of view, it is even more important to known that in the event of an emergency situation, employed tugs are capable of providing the assistance necessary in order to avoid an accident [4, 5, 6]. Throughout the 1970s and 1980s, the state-of-the-art began to describe the tugs' capabilities basing on polar diagrams of thrust vectors [7]. However, polar diagrams have two important handicaps to make an accurate comparison between tugs: they only represent tug capabilities at zero speed, and thrust capabilities are represented in a coordinate system referenced to the tugs themselves instead of being referenced to assisted vessels [8]. For this reason, although it is valid to analyse the abilities of tugs at very low speeds, they are not useful when assistance are carried out with an important forward speed. To overcome this lack of objective information from polar diagrams, in 2001, for the first time, researchers represented the forces generated by a purpose-built escort tug referred to as the Cartesian coordinate system of the assisted vessels [9]. This new diagram is known as a \"butterfly\" diagram, and through introducing the speed and the \"appearance\" of the tug, it is possible to read magnitudes of transverse (steering) and longitudinal (braking) forces generated on the assisted vessel at 360\({}^{\circ}\) and at different speeds. However, \"butterfly\" diagrams do not specify, for example, the displacement and the manoeuvrability of the escorted vessel. Therefore, forces generated by an escort tug can be sufficient to assist a particular vessel, but not to another, which would seriously jeopardise the overall safety of the operation. On the other hand, BP only can be used to differentiate the capabilities at zero speed, because it is defined as the maximum force (thrust) exerted by the tug on a fixed towline in static condition [10; 11; 12]. Nowadays, this is the main parameter used to assess the capabilities of ship-handling tugs working at lower speeds, but if tugs have a certain speed over the water, BP will be different with respect to the static condition, mainly because at each speed, the hydrodynamic resistance consumes different energy. This circumstance was more evident with the implementation of escorting assistance after the accident of \(m/v\) \"Exvon Valdez\" in 1989 and the consequent promulgation of the Oil Pollution Act in 1990 (OPA 90) [13; 14; 15]. It became clear that BP was not a good tool to assess the tugs' capabilities during escorting assistance sailing at high speeds (above 6 knots) over long distances [16]. In these operations, the hydrodynamic lift from the tug's hull will be used to develop a large force for transmitting through the rope into the tow [17], which is proportional to the tug's hull underwater area and the square of transit speed [18; 19]. Therefore, the performance of escort tugs must be defined, at least, by the capability to generate steering forces using indirect methods up to 10 knots [20; 21]. This force, known as Tons of Steering Pull (TSP), is more important and higher than BP, and it corresponds to a steering force used to alter the vessel's course in order to head it to a safe area; to help it to reduce the forward speed; or to counteract the effects of its rudder locked into a side, sailing at 10 knots [22; 23; 24]. Despite this, for many past decades and even nowadays, an important portion of the towing industry consider BP as the most important and only parameter that defines the characteristics of all types of tugs [4; 25].