## 1 Introduction  
Race car drivers are expected to achieve a minimum lap time by exploiting all of the grip potential of a given vehicle [1; 2; 3]. Given a driving track, the driver should make optimal use of the available control inputs, mainly including the accelerator and brake pedal positions, along with the steering wheel angle. Minimum lap time strategies have been investigated for more than 60 years [4]. They are very challenging problems, but with potentially very wide applications, including the analysis of race tracks (e.g., Formula 1) from the point of view of the human drivers and track engineers, the optimisation of vehicle design/setup, and the potential combination of advanced vehicle control techniques [5]. The most common methods for lap time simulation are [6]: (i) optimal control problem (OCP) approaches; and (ii) quasi-steady-state (QSS) approaches. Optimal control is used when the trajectory is not known a priori. This might be the case when it is necessary to find the optimal trajectory, e.g., the first time a track is studied or simply when the track is new, as it sometimes happens, even in Formula 1. OCPs are approachable either with direct methods (which solve the problem using nonlinear programming (NLP) techniques [7; 8]) or with indirect methods (which solve the problem using numerical techniques for systems of differential equations [9; 10]). The key peculiarities of the direct and indirect methods have been extensively discussed in the literature [11]. QSS methods, instead, rely on a given trajectory. They are based on steady-state vehicle dynamics equations, with the exception of the vehicle speed. Normally, the path is divided into small segments and the corner apexes are identified; hence, the maximum vehicle speed is therein calculated. The acceleration and braking zones are worked out by forward and backward integration, respectively, after and before each apex, and the speed profiles between the apexes are connected at their intersection points [12]. While integrating backwards, the maximum braking capabilities are considered, which normally include grip and aerodynamic effects. On the other hand, forward integration accounts for the maximum acceleration capabilities, which may include grip, aerodynamics, and power limits. In the literature [13], a method is proposed that employs QSS models and free trajectory using a vehicle g-g map, and then applying the OCP techniques again. Further simulation studies have been proposed to investigate additional aspects, including the extension of QSS methods to transients [14; 15] and modelling three-dimensional roads [16]. In terms of computational cost, both OCP and QSS are generally quite demanding. This is because of the solution of differential equations through direct/indirect methods (OCP) and the backward/forward integration from each apex (QSS), respectively. In the analysis of OCP techniques in reference [11], the authors suggested that new methods should be investigated that \"have the ability to compute a simulation in a reasonable amount of time, e.g., less than a few hours\". For example, the OCP method in reference [17] took around 8 hours to simulate a full lap of the Jerez circuit. QSS methods are generally faster than the OCP ones. The QSS method in reference [12] took 16 minutes to simulate the Barcelona circuit, compared with 39.8 hours using the OCP method in reference [18]. Still, that is around 5-10 times (depending on the vehicle) the actual lap time. In addition, the authors of the QSS technique in reference [19] stated that their method is \"for design phase decisions but not for a fast trackside usage\". In this paper, a new QSS approach is proposed for fixed trajectories. It is based on a numerical method using a simplified vehicle model including physical constraints, such as combined tyre-road interaction limits, aerodynamic effects, and engine power boundaries. The main advantage of the proposed method lies in the low computational cost.