Minimum Lap Time Research Articles

Determination and description of a vehicle's maximum jerk capacity

Both Transient Optimal and Quasi-Steady-State vehicle modelling are frequently used for minimum time manoeuvre and minimum lap-time simulations for motorsport/vehicle development purposes. Quasi-Steady-State methods have been shown to perform such tasks with low computational cost but produce results with measurable deviation from equivalent Transient Optimal studies. It is hypothesised that bounding point-mass models with jerk limits will improve alignment with Transient Optimal simulations through the consideration of transient behaviours such as control application rates and inertia. This work presents a new method for the determination of a vehicle's maximum jerk capacity from a seven degree-of-freedom vehicle model. A range of results from this proposed technique are presented and show the sensitivity of jerk capacity to changes in control application rate limits and yaw inertia, as well as current vehicle velocity, acceleration, and jerk. It is proposed that further exploration of the validity of the jerk limit calculation method take place through characterisation of said limits and application to a point-mass model, after which comparison to Transient Optimal and Quasi-Steady-State equivalents can occur.

Schwarz decomposition for parallel minimum lap-time problems: evaluating against ADMM

The Minimum Lap Time Problem (MLTP) remains a significant area of research, particularly in the motorsport context. This form of Optimal Control Problem (OCP) aims to minimise lap times on a specific track with a given vehicle. Various complexities in both vehicle and track models are employed across the literature to address optimal trajectory planning. While previous works have tackled MLTP as a singular task using a serial approach, the increasing model complexity and horizon length demands the utilisation of parallelisation techniques. This paper introduces a novel application of the Overlapping Schwarz Decomposition algorithm to address the MLTP. The algorithm divides the problem into smaller sub-problems based on different sectors of a track, distributing them among multiple processors. We validate and compare the Schwarz approach against a serial approach and the Alternating Direction Method of Multipliers (ADMM) in solving MLTP with over 2.5 million variables. Despite the general efficiency improvement of parallelisation compared to the serial approach, the Schwarz algorithm demonstrates superior speed, accuracy and robustness compared to ADMM. As a result of our findings, it emerges as the preferred choice when large-scale MLTPs need to be solved.

Tyre and brake thermal management in NASCAR

The need to manage race car tyres is routine faire and several academic papers have already appeared on this topic. We contribute to this literature with refined tyre grip and wear-rate models. Less well known is a methodology for managing the thermal transients in the car's braking system. This knowledge gap is addressed with a physically-based disc-and-calliper brake-heating model. The parameters in each of the models are optimised against measurement data captured on an instrumented Generation 7 (Gen 7) NASCAR driven on the Darlington speedway. These models are replicated on each corner of the vehicle. The remainder of the paper is directed to studying the minimum lap time performance of a Gen 7 NASCAR that is subject to tyre wear and temperature variations; brake heating influences are emphasised. Due to their novelty, the results focus on the performance impact of temperature variations and constraints on the front brake disc temperatures. Constraints on the brake temperatures are introduced to protect the brakes from fade and thermally-induced damage.

Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle

Motorcycle motion is largely influenced by the road geometry, which alters the allowable accelerations in longitudinal and lateral directions and influences the vertical wheel loads. Recently, a method for three-dimensional road reconstruction and its incorporation into transient and quasi-steady-state (QSS) minimum lap time simulations (MLTSs) has been proposed. The main purpose of this work is to demonstrate how significantly different results from a minimum lap time optimal control problem can be obtained when using inappropriate elevation data sources in the track reconstruction problem, and how the road model reconstructed using poor input data can lead to misleading conclusions when analyzing real vehicle and driver performances. Two road models derived from high- and low-resolution digital elevation models (DEMs) are compared and their impact on the optimal maneuver of a racing motorcycle is examined. The essentials of track identification are presented, as well as vehicle positioning on the 3D road and the generalized QSS motorcycle model. Obtained 3D and 2D road models are analyzed in detail, on a case example of the Road Atlanta racetrack, and used in minimum lap time simulations, which are validated by the experimental data recorded on the Supersport motorcycle. The comparative analysis showed that great care should be taken when selecting the elevation dataset in the track reconstruction process, and that the 1 m resolution local DEMs seem to be sufficient to obtain MLTS results close to the measured ones. The example of using the 3D free-trajectory QSS minimum lap time problem to localize the track segments where real driver actions can be improved is also presented. The differences between simulation results on different road models of the same racetrack can be large and influence the interpretation of optimal maneuver.

Minimum-Lap-Time Planning of Multibody Vehicle Models via the Articulated-Body Algorithm

Minimum lap-time planning (MLTP) is a well-established problem in the race car industry to provide guidelines for drivers and optimize the vehicle’s setup. In this paper, we tackle the 3D nature of the problem in its full extension, making no simplifying assumptions on the mechanics of the system. We propose a multibody vehicle model, described by rigorous dynamical equations. To effectively handle the resulting complexity, we devised an efficient direct dynamics computational method based on Featherstone’s articulated-body algorithm (ABA). To solve the MLTP, we employed a direct-collocation technique, discretizing the problem so that all information of the 3D track is pre-processed and directly embedded into the discrete problem. This discretization approach turns out to be perfectly compatible with our vehicle model, leading to a solution in accessible computational time frames. The high level of detail of the model makes the proposed approach most useful for in-depth vehicle dynamics analyses on complex tracks. To substantiate the analysis, we provide a comparison with the results obtained by a double-track model on the Nürburgring Nordschleife circuit. Consistently with the average trend defined by the double track, the proposed model features a more dynamically rich behavior, realistically capturing the higher-order effects elicited by the sharp corners and the highly variable slope of the track.

A quasi-steady-state minimum lap time simulation of race motorcycles using experimental data

In this work, the minimum-lap-time problem is investigated using a quasi-steady-state model. The prior research approach has been modified by introducing experimentally based controls’ constraints to consider some effects of vehicle inertia and human physical limitations. The differences between the theoretical and recorded experimentally g-g diagrams are analysed, and a semi-empirical g-g diagram is proposed as an approximation for the usage of most motorcycle riders. The effect of the braking technique (optimum and using front brake only) is discussed on a variable radius ‘U’ trajectory example. A new system of OCP state equations is proposed, and optimal trajectories are computed for several test scenarios for open and closed-loop tracks. The simulation result for the closed-loop track is compared with the experimental data from a GPS-based data acquisition system. The standard and proposed approach are compared, and conclusions are drawn.

Optimization-based path planning and collision avoidance for autonomous racing

This paper presents a hierarchical motion planner for autonomous racing. The long-term motion planner functions offline and formulates the optimal motion plan for the entire race track. The short-term collision avoidance planner functions online and formulates a motion plan for a limited horizon ahead of the autonomous car when an obstacle is detected in the path of the vehicle. The motion planners formulate the planning problems as optimal control problems and solve the resulting optimizations using an interior point optimizer (IPOPT). Simulation experiments show that an autonomous vehicle using the motion planner is able to race around the track with minimum lap time while avoiding unexpected obstacles.

Double-Iterative Gaussian Process Regression for Modeling Error Compensation in Autonomous Racing

Autonomous racing control is a challenging research problem as vehicles are pushed to their limits of handling to achieve an optimal lap time; therefore, vehicles exhibit highly nonlinear and complex dynamics. Difficult-to-model effects, such as drifting, aerodynamics, chassis weight transfer, and suspension can lead to infeasible and suboptimal trajectories. While offline planning allows optimizing a full reference trajectory for the minimum lap time objective, such modeling discrepancies are particularly detrimental when using offline planning, as planning model errors compound with controller modeling errors. Gaussian Process Regression (GPR) can compensate for modeling errors. However, previous works primarily focus on modeling error in real-time control without consideration for how the model used in offline planning can affect the overall performance. In this work, we propose a double-GPR error compensation algorithm to reduce model uncertainties; specifically, we compensate both the planner's model and controller's model with two respective GPR-based error compensation functions. Furthermore, we design an iterative framework to re-collect error-rich data using the racing control system. We test our method in the high-fidelity racing simulator Gran Turismo Sport (GTS); we find that our iterative, double-GPR compensation functions improve racing performance and iteration stability in comparison to a single compensation function applied merely for real-time control.

Convex Performance Envelope for Minimum Lap Time Energy Management of Race Cars

The optimization of the energy management of modern hybrid-electric or fully electric race cars for minimum lap time requires a description of the vehicle dynamics performance envelope, that is, of the tires’ grip limit in corners, braking zones and during acceleration. In this paper, we present a computationally efficient performance envelope model in the form of convex constraints on the achievable longitudinal and lateral acceleration, on the assumption that the path on the track is given. The proposed acceleration limits are modeled velocity-dependent to take into account the effect of aerodynamic downforce present in many circuit race cars. The formulation as linear equality, inequality and second-order cone constraints allows to embed the model in a convex energy management optimization framework. To showcase the approach, we identify the model with data obtained from a state-of-the-art hybrid-electric Formula 1 car and present results for the Silverstone and Spa-Francorchamps circuits. The optimal energy management strategies can be evaluated with a computational time of less than 1 s. The optimal velocity profile subject to the performance envelope constraints is close to the measured one. The good agreement between the optimal solution and the measurement data shows that the proposed model captures the vehicle dynamics accurately enough for the purposes of energy management optimization.

Minimum lap time trajectory optimisation of performance vehicles with four-wheel drive and active aerodynamic control

ABSTRACT As racing vehicles become more complex, optimising the interaction between subsystems becomes critical for racing performance. In this work, we incorporate two such subsystems into a vehicle model. We investigate the performance benefits of a four-wheel-drive vehicle with independent control over its in-hub motors and active control on the rear wing flap rotation. The performance is evaluated by solving minimum lap time optimal control problems (OCP) for various vehicle configurations. The OCP is transformed into a nonlinear programming problem through direct collocation and is solved by an interior point method. The four-wheel-drive configuration performs better than rear-wheel drive in terms of lap time, finishing the Barcelona circuit faster. The benefits come mainly from higher longitudinal accelerations. Active aerodynamic control improves performance regardless of the propulsion configuration, leading to another of lap time improvement on the Barcelona circuit. The optimal aerodynamic control strategy is different between propulsion configurations, particularly on corner exits. This model enables the exploration of optimal torque vectoring controls and their interaction with a vehicle's active aerodynamics. This is needed to guide the design of the increasingly complex racing vehicle controllers of the future.

Optimal recovery manoeuvres of racing motorcycles

The focus of this work is on the optimal recovery manoeuvres of racing vehicles, which are here defined as the manoeuvres that need be performed in order to complete the lap in the minimum time, after a driving error occurs. The driving errors considered are speed excess and trajectory deviations, with respect to the baseline conditions. The baseline conditions are those giving the minimum lap time, and are vehicle and track dependent. The practical applications of the methodology proposed include the estimation of the driving errors that can be recovered while remaining within the racetrack borders, and the estimation of the minimum asphalted run-off area that need be provided to recover from driving errors of given magnitude. An example of application is given for a racing motorcycle on the Mugello circuit.

Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles

Minimum-lap-time simulations with quasi-steady-state models and predefined trajectory have been in use for many years. However, most of the published works deal with two-dimensional roads and employ the so-called apex-finding method. This paper focuses on the application to three-dimensional roads, both with an extension of the ‘apex-finding’ approach to three-dimensional scenarios and with an optimal control method. Both approaches are based on g–g–g diagrams, i.e. the three-dimensional extension of the well-known g–g maps. In addition, under the assumption that the predefined trajectory is determined from noisy data (e.g. logged from the real vehicle), the three-dimensional trajectory reconstruction problem is addressed, to find a smooth and drift-free racing line to be used in the minimum-time simulation – again an optimal control approach is employed for the trajectory reconstruction. Examples of application are given both for a car and a motorcycle.

Curved-ribbon-based track modelling for minimum lap-time optimisation

Three-dimensional road models for vehicular minimum-lap-time manoeuvring are typically based on curvilinear coordinates and generalizations of the Frenet–Serret formulae. These models describe the road as a parametrized ‘ribbon’, which can be described in terms of three curvature variables. In this abstraction the road is assumed laterally flat. While this class of road models is appropriate in many situations, this is not always the case. In this research we extend the laterally-flat ribbon-type road model to include lateral curvature. This accommodates the case in which the road camber can change laterally across the track. Lateral-position-dependent camber is introduced as a generalisation that is required for some race tracks. A race track model with lateral curvature is constructed using high-resolution LiDAR measurement data. These ideas are demonstrated on a NASCAR raceway, which is characterized by large changes in lateral camber angle (approx 10^circ) on some parts of the track. A free-trajectory optimization is employed to solve a minimum-lap-time optimal control problem. The calculations highlight the practically observed importance of lateral camber variations.

Analysis of driving styles of a GP2 car via minimum lap-time direct trajectory optimization

This paper addresses the problem of the link between the driving style of an ideal driver, modelled as an optimal controller, and fundamental set-up parameters of a vehicle in the GP2 motorsport class. The aim is to evaluate quantitatively how set-up parameters, like distribution of aerodynamic loads, weight and roll stiffness between front and rear axles, affect the driving style, encoded in the shape of the optimal trajectory and in the acceleration, brake and steer inputs. To this aim, we develop an optimization code that includes a double-track vehicle model capable of solving the minimum lap-time problem (MLTP) on a given track. The track is represented via NURBS curves and the MLTP is framed and solved as an optimal control problem by transcription into a nonlinear program using direct collocation. To assess the accuracy of the vehicle model and the optimization pipeline, we also validate our results against real telemetry data. The developed software framework lends itself to easily perform both sensitivity analysis and concurrent trajectory planning and set-up parameter optimization: this is obtained by simple promotion of static parameters of interests to variables in the optimal control problem. Some results along these lines are also included.

A Convex Optimization Framework for Minimum Lap Time Design and Control of Electric Race Cars

Lap time is the most relevant performance indicator in electric racing. In order to minimize it, it is important to carefully design the powertrain of the race car in terms of sizing and technology, and meticulously administer the energy available on-board. To facilitate efficient design, this paper presents a convex optimization framework to rapidly compute the minimum-lap-time control strategies for a battery electric race car. Specifically, we first identify a convex model of the electric powertrain, including the battery, the electric motor, and two transmission technologies: a fixed-gear transmission (FGT) and a continuously variable transmission (CVT). Second, assuming an expert driver, we formulate the minimum-lap-time control problem in a convex fashion for a given driving path, and compute its globally optimal solution with second-order conic programming algorithms. Third, we showcase our framework on the Le Mans track by comparing the performance achievable with an FGT and a CVT, and validate it with nonlinear simulations. Our results show that for the given setup a CVT can balance its lower efficiency and higher weight with a higher-efficiency and more aggressive motor operation, significantly outperforming the lap time achievable with an FGT. Finally, we leverage the computational efficiency of our framework to carry out parameter studies on the components, revealing that optimizing the size of the battery and the motor for the specific scenario can considerably improve the achievable lap time, and that the best transmission strongly depends on the sizing decisions.

Time-optimal gearshift and energy management strategies for a hybrid electric race car

Modern Formula 1 race cars are hybrid electric vehicles equipped with an internal combustion engine and an electric energy recovery system. In order to achieve the fastest possible lap time, the components’ operation must be carefully optimized, and the energy management must account for the impact of the gearshift strategy on the overall performance. This paper presents an algorithm to calculate the time-optimal energy management and gearshift strategies for the Formula 1 race car. First, we leverage a convex modeling approach to formulate a mathematical description of the powertrain including the gearbox, preserving convexity for a given engine speed trajectory. Second, we devise a computationally efficient algorithm to compute the energy management and gearshift strategies for minimum lap time, under consideration of given fuel and battery consumption targets. In particular, we combine convex optimization, dynamic programming and Pontryagin’s minimum principle in an iterative scheme to solve the arising mixed-integer optimization problem. We showcase our algorithm with a case study for the Bahrain racetrack, underlining the interactions between energy management and gear selection. Finally, we use our approach as a benchmark to evaluate the sub-optimality of a heuristic gearshift rule. Our results show that using an optimized engine speed threshold for upshifts can yield close-to-optimal results. However, already deviations smaller than 4% from the best possible threshold can increase lap time by more than 100ms, highlighting the importance of jointly optimizing energy management and gearshift strategies.

Brachistochrone on a velodrome

The Brachistochrone problem, which describes the curve that carries a particle under gravity in a vertical plane from one height to another in the shortest time, is one of the most famous studies in classical physics. There is a similar problem in track cycling, where a cyclist aims to find the trajectory on the curved sloping surface of a velodrome that results in the minimum lap time. In this paper, we extend the classical Brachistochrone problem to find the optimum cycling trajectory in a velodrome, treating the cyclist as an active particle. Starting with two canonical cases of cycling on a sloping plane and a cone, where analytical solutions are found, we then solve the problem numerically on the reconstructed surface of the velodrome in Montigny le Bretonneux, France. Finally, we discuss the parameters of the problem and the effects of fatigue.

Optimal Tyre Management of a Formula One car

Optimal control calculations are used to study the effect of tyre wear on race car performance. This is achieved by solving a minimum lap time optimal control problem over multiple laps using a dynamical model of a Formula One car. A previously developed thermodynamic model is enhanced by adding an additional state for the carcass temperature of the tyres. The tyre grip is modelled as a function of the tyres’ wear and temperature. Grip reduces when tyres get worn, or the tyres are not operated within their optimal temperature window. Overheating the tyres can accelerate wear, which in turn, degrades performance. The optimal control problem solver needs to ‘manage’ the state of tyres throughout a race (not just a single lap) to ensure that optimal race performance is achieved. At some point during a race a pit stop may be required to change worn tyres so that tyre grip can be restored. It is essential to understand the wear characteristics of various tyre compounds in order to determine the point when the time needed for a pit stop is justified in terms of subsequent racing performance.

Time-optimal trajectory planning for a race car considering variable tyre-road friction coefficients

This paper shows the planning of time-optimal trajectories, which allows an autonomous race car to drive at the handling limits, taking into account locally changing road friction values. For this purpose, the minimum lap time problem is described as an optimal control problem, converted to a nonlinear programme using direct orthogonal Gauss-Legendre collocation and then solved by the interior-point method IPOPT. Reduced computing times are achieved using a curvilinear abscissa approach for track description, algorithmic differentiation using the software framework CasADi, and a smoothing of the track input data by approximate spline regression. The vehicle's behaviour is approximated as a single track and double track model with quasi-steady state tyre load simplification and nonlinear tyre model. The results are used to evaluate which vehicle physics are important for the calculation of the time-optimal trajectory. The novelty of this work is the consideration of wheel-specific tyre-road friction coefficients along the racetrack using a track friction map. It is shown that variable friction coefficients have a significant impact on the trajectory, and therefore significantly improve lap times on inhomogenous racetracks. The proposed trajectory planning has proven its practical suitability in first tests on an autonomous race car and will be used in the coming racing season in the Roborace competition.

Minimum Lap Time Control of Hybrid Electric Race Cars in Qualifying Scenarios

This paper presents models and control algorithms for the energy management of the Formula 1 race car in qualifying scenarios, whereby the car's performance is solely determined by the lap time achieved over a single lap. Specifically, we first extend an existing convex model of the Formula 1 car to include the waste-gate action, and analytically derive its lap-time-optimal control policy. Second, we combine nonlinear programming techniques with convex optimization to numerically compute the optimal solution for an out-lap-flying-lap sequence. Third, we devise equivalent lap time minimization strategies, whereby three PI controllers adapt the optimal control policy implemented on the car in real time, in order to track precomputed control strategies in a minimum-lap-time fashion. Finally, we benchmark the control algorithms presented against the optimal solution and validate their performance with high-fidelity simulations. The results obtained show the optimality and the robustness of the presented approach.