FSAE Lap Time Simulation Vehicle Model Development and Optimization

5.1.1 Simulation Structure Overview

LTS26 comprises over 3,000 lines of code implemented across MATLAB and Simulink, incorporating both a transient model and a quasi-steady-state model to address the distinct requirements of different competition event types. The tool produces comprehensive performance outputs for all dynamic competition events, while achieving a reduction in computation time of more than half relative to the preceding LTS25 tool.

The simulation programme is structured into two primary components: the Vehicle Model and the Dynamic Model. The Vehicle Model defines the mathematical representation of the race car's dynamics, consisting of a set of equations that describe the physical behaviour of the system. The Dynamic Model, in contrast, contains the algorithms that govern the execution of the simulation. During operation, the Dynamic Model repeatedly invokes the Vehicle Model to evaluate vehicle performance under varying conditions. In this architecture, the Vehicle Model serves as the computational engine, while the Dynamic Model governs the conditions under which this engine is exercised.

5.1.2 Revamped Structure

A primary contributor to the computational inefficiency of LTS25 was that its processing time scaled directly with the number of discretisation points in the track model. At each mesh point, the simulation independently evaluated the vehicle's cornering, acceleration, and braking capabilities, with the output updated point by point as the computation progressed.

To address this fundamental inefficiency, the simulation framework in LTS26 decouples vehicle performance evaluation from the track representation entirely. Rather than performing point-by-point calculations along the track, the simulation first constructs a comprehensive representation of the vehicle's performance, the Performance Envelope. This envelope characterises the vehicle's maximum capabilities across the full range of operating conditions.

Since vehicle performance varies with speed and other operating parameters, the envelope is constructed by evaluating the vehicle model at discrete operating points, followed by interpolation to generalise the results across the continuous operating domain. Once established, this Performance Envelope can be applied to any track geometry without repeating the underlying vehicle model computations, substantially reducing the total computational cost and significantly improving the flexibility and reusability of the simulation.

5.1.3 Course Differentiation

In the Acceleration and Skidpad events, the vehicle operates predominantly under steady-state conditions that is relatively easier to model comparing to the Autocross and Endurance events, which involve continuous and simultaneous variation of both longitudinal and lateral accelerations, representing a considerably more complex dynamic environment.

Given that each event type presents unique modelling requirements, the simulation framework in LTS26 was further differentiated beyond the structural overhaul described above. Specifically, the Acceleration and Braking simulations are implemented using a transient model in Simulink, Skidpad to be simulated using a steady-state model, and the Autocross and Endurance simulations are implemented using a quasi-steady-state model in MATLAB scripts. This separation improves both simulation accuracy and numerical stability.

5.2.1 Acceleration Simulation

During acceleration, the performance of the race car is governed primarily by two competing limiting factors: tyre traction forces and powertrain driving forces. When the torque delivered by the powertrain exceeds the maximum traction force available at the driven tyres, wheel spin occurs and tyre grip becomes the binding constraint. Conversely, when tyre grip exceeds the available powertrain output, acceleration becomes power-limited. Accurate prediction of the spatial distribution of these two limiting regimes along the acceleration track is therefore essential for optimising vehicle design parameters and improving event performance.

As the vehicle accelerates, its instantaneous speed increases continuously, which in turn affects both the motor torque output and the aerodynamic forces acting on the vehicle, which vary with the square of velocity. For this reason, vehicle speed is selected as the primary feedback variable within the simulation loop.

5.2.2 Braking Simulation

During braking, vehicle performance is similarly governed by the interaction between tyre forces and the applied braking force. However, since the braking force generated by the hydraulic system can be considered sufficiently large relative to the tyre's capacity to transmit longitudinal force, it is treated as effectively unlimited within this model. Under this assumption, braking performance is simplified to be tyre-limited throughout the event.

As with the acceleration simulation, vehicle speed is adopted as the primary feedback variable, as it directly governs both the aerodynamic loading and the tyre behaviour at each instant during the braking event.

5.3.1 Vehicle Model

During cornering, the centripetal forces required to sustain the vehicle's curved motion are generated collectively by the four tyres, each contributing forces of varying magnitude and direction depending on the prevailing operating conditions. The tyre force at each corner is a function of slip angle, slip ratio, normal load, inclination angle, and tyre-specific parameters characterised from experimental data.

\[F_{\text{Tire}} = {f}(\text{Slip Angle},\ \text{Slip Ratio},\ \text{Tire Load},\ \text{Inclination Angle},\ \text{Tire Parameters})\tag{5.1}\]

To determine the resultant tyre forces, several intermediate quantities must be evaluated. Among the most critical is the slip angle, which governs the generation of lateral tyre force and is defined based on the decomposition of the wheel's velocity vector into longitudinal and lateral components relative to the tyre contact patch.

Slip angle is defined based on the decomposition of the velocity vector into longitudinal and lateral components relative to the tire.

\[\alpha_{FR} = -\delta + \arctan\!\left(\frac{V_y + \omega_r \cdot a}{V_x + \omega_r \cdot d/2}\right) \tag{5.2}\] \[\alpha_{FL} = -\delta + \arctan\!\left(\frac{V_y + \omega_r \cdot a}{V_x - \omega_r \cdot d/2}\right) \tag{5.3}\] \[\alpha_{RR} = \arctan\!\left(\frac{V_y - \omega_r \cdot b}{V_x + \omega_r \cdot d/2}\right) \tag{5.4}\] \[\alpha_{RL} = \arctan\!\left(\frac{V_y - \omega_r \cdot b}{V_x - \omega_r \cdot d/2}\right) \tag{5.5}\]

where \( \delta \) is steering angle, \( \omega_r \) is vehicle's yaw rate, \( a \) is distance from vehicle's center of gravity (CG) to the front axle, \( b \) is distance from vehicle's CG to the rear axle, \( d \) is vehicle's track.

Decomposed velocity vector, \( V_x \) and \( V_y \) are derived from the vehicle's speed and body slip angle \( \beta \):

\[V_x = V \cdot \cos(\beta) \tag{5.6}\] \[V_y = V \cdot \sin(\beta) \tag{5.7}\]

A second critical input to the tyre force model is the normal load at each corner, which arises from four contributing sources.

\[F_z = \frac{\text{Mass} \times 9.81}{4} \tag{5.8}\] \[F_{\text{Aero}} = \frac{1}{4} \times \left(0.5 \times \text{Density} \times CL \times \text{Area} \times V^2\right) \tag{5.9}\] \[F_{\text{Lat_LT}} = F_z \times \frac{A_y \times H_{\text{CG}}}{9.81 \times d} \tag{5.10}\] \[F_{\text{Long_LT}} = F_z \times \frac{A_x \times H_{\text{CG}}}{9.81 \times \text{wheelbase}} \tag{5.11}\]

Where, Mass is the vehicle's total mass, Density is the density of fluid (in this case, air), \( CL \) is the Coefficient of Lift, Area is the vehicle's frontal area, \( V \) is the vehicle's speed value, \( A_y \) is the vehicle's Lateral Acceleration, \( A_x \) is the vehicle's Longitudinal Acceleration, \( H_{\text{CG}} \) is the height of the vehicle's CG, and \( d \) is the vehicle's track length. Here we assume that the vehicle has equally distributed static weight and aerodynamic downforce across four wheels.

Assuming the vehicle is negotiating a left-hand turn while accelerating, the resulting normal load at the rear-right tyre is accordingly given by:

\[F_{zrr} = F_z + F_{\text{Aero}} + F_{\text{Lat_LT}} + F_{\text{Long_LT}} \tag{5.12}\]

Sign conventions are adjusted accordingly for the remaining wheel positions. Tyre inclination angle is assumed to be zero in the current model, given its relatively minor contribution to tyre force generation at this level of modelling fidelity.

Tyre characteristics are defined using pre-fitted parameters derived from experimental measurements and are treated as fixed known inputs to the optimisation problem.

Upon evaluating the longitudinal and lateral forces at each of the four tyres, the complete four-wheel vehicle dynamic geometry is referenced to compute the resultant longitudinal force, lateral force, and yaw moment with respect to the fixed vehicle coordinate frame.

\[\begin{aligned}F_y &= F_{yFR} \cdot \cos(\delta) + F_{yFL} \cdot \cos(\delta) + F_{xFR} \cdot \sin(\delta) + F_{xFL} \cdot \sin(\delta) \\ &\quad + F_{yRL} + F_{yRR}\end{aligned} \tag{5.13}\] \[\begin{aligned}F_x &= F_{xFR} \cdot \cos(\delta) + F_{xFL} \cdot \cos(\delta) - F_{yFR} \cdot \sin(\delta) - F_{yFL} \cdot \sin(\delta) \\ &\quad + F_{xRL} + F_{xRR}\end{aligned} \tag{5.14}\] \[\begin{aligned}M_z &= a \cdot \big(F_{xFR} \cdot \sin(\delta) + F_{xFL} \cdot \sin(\delta) + F_{yFR} \cdot \cos(\delta) + F_{yFL} \cdot \cos(\delta)\big) \\ &\quad - b \cdot \big(F_{yRL} + F_{yRR}\big) \\ &\quad + \frac{d}{2} \cdot \big(F_{xFR} \cdot \cos(\delta) - F_{xFL} \cdot \cos(\delta)\big) \\ &\quad + \frac{d}{2} \cdot \big(F_{xRR} - F_{xRL}\big) \\ &\quad + \frac{d}{2} \cdot \big(F_{yFL} \cdot \sin(\delta) - F_{yFR} \cdot \sin(\delta)\big)\end{aligned} \tag{5.15}\]

The vehicle accelerations with respect to the path-tangential coordinate frame are then given by:

\[a_x = \frac{1}{m} \cdot \Big(F_y \sin(\beta) + F_x \cos(\beta) - \text{Drag}\Big) \tag{5.16}\] \[a_y = \frac{1}{m} \cdot \Big(F_y \cos(\beta) - F_x \sin(\beta)\Big) \tag{5.17}\]

At this stage, several variables remain undetermined, specifically the steering angle \( \delta \), body slip angle \( \beta \), yaw rate \( \omega_r \), and the slip ratios at each tyre. These quantities are treated as decision variables within the optimisation problem. To ensure internal consistency between the assumed and computed accelerations, residual equality constraints are introduced:

\[A_{x_{Res}} = A_x - A_{x_{in}} \tag{5.18}\] \[A_{y_{Res}} = A_y - A_{y_{in}} \tag{5.19}\]

5.3.2 Dynamic Model

5.3.2.1 Steady-State Model

As the goal being constructing a G-G diagram that captures the vehicle's maximum potentials, combinations of longitudinal and lateral accelerations values need to be output from simulation to cover all possible scenarios during dynamic events. This will be achieved by performing numerical optimization using the Vehicle Model formulated above with specific set of decision variables.

A fundamental consideration in the design of the steady-state model is that numerical optimisation, by formulation, permits only a single objective variable to be minimised or maximised per solver call, and the remaining unknowns are determined as a consequence of satisfying the imposed constraints. The core algorithmic challenge is therefore to design a structured approach that ensures both A_x and A_y are simultaneously driven towards their maximum achievable values, despite the solver being capable of directly optimising only one at a time.

The first design iteration addressed this by decomposing the problem into a sequential series of optimisation calls. Since tyre grip is most efficiently utilised when directed along a single axis, the maximum and minimum achievable longitudinal accelerations are first determined in isolation by setting lateral acceleration to zero. These two optimisation problems define the feasible range of A_x values. The programme subsequently generates a uniform mesh of A_x values between these bounds, and for each discrete value imposes A_x as a fixed constraint while maximising A_y as the objective. This yields a set of optimised (A_x, A_y) pairs that collectively trace the G-G boundary.

While conceptually straightforward, this formulation required a minimum of three separate nonlinear optimisation problem configurations, increasing implementation complexity. More critically, the rectangular mesh imposed on A_x did not align naturally with the circular geometry of the G-G boundary, resulting in query points near the pure braking and pure acceleration extremes being sparsely distributed relative to the combined acceleration region. This mismatch caused a high rate of optimisation failure at the boundary regions where the feasible set is most constrained, and produced uneven resolution across the G-G diagram.

To address these shortcomings, a revised formulation was adopted in which the optimisation objective is no longer defined in terms of either longitudinal or lateral acceleration independently, but instead in terms of a scalar grip factor P, defined as:

\[\left| P \right| = \sqrt{A_x^2 + A_y^2} \tag{5.20}\]

The programme defines an angular sweep variable θ uniformly distributed over [−π, π]. This polar formulation naturally distributes query points evenly along the G-G boundary, reduces the problem to a single unified optimisation structure, and eliminates the need to pre-determine the A_x bounds. The angular mesh increment was selected by balancing diagram resolution against computational cost, and initial guesses at each instance are warm-started from the preceding solution to exploit the expected smoothness of the boundary and improve convergence reliability.

A full description of the numerical optimization setup can be found in Appendix A.6.

5.3.2.2 Quasi-Steady-State Model

To simulate the complete performance of a race car competing in events such as Autocross and Endurance, the Steady-State Model is repeated systematically across the full operational speed range of the vehicle. This procedure constructs a G-G-V diagram that characterises the vehicle's dynamic limits across all operating conditions.

The G-G diagram computations were performed at speeds ranging from 10 m/s up to the vehicle's theoretical maximum speed, with a uniform increment of 1 m/s. Speeds below 10 m/s were excluded from direct computation since it's uncommon to occur in FSAE competition, and the vehicle has significant different behaviours at such low speeds. To account for this excluded regime, linear extrapolation was applied to the computed results immediately above the 10 m/s boundary to estimate the vehicle's performance below this speed.

5.3.2.3 Dynamic Event Simulation

With the vehicle's performance limits established through the G-G-V diagram, it becomes possible to predict the vehicle's dynamic capabilities on an actual competition track, as encountered during the Autocross and Endurance events. The simulation proceeds in two stages: first, a Boundary Speed Profile (BSP) is generated to represent the maximum achievable cornering speed at each point along the track; second, the maximum achievable speed accounting for the constraints imposed by acceleration and braking is computed, yielding the Limit Speed Profile (LSP).

To construct the BSP, the programme determines the maximum achievable lateral acceleration A_y at each speed value from the interpolated G-G-V diagram, representing the vehicle's peak cornering capability at that speed. The minimum achievable turn radius for each speed-acceleration combination is then computed as:

\[R = \frac{V^2}{A_y} \tag{5.21}\]

The discrete results are subsequently interpolated to generalise the relationship across all cornering scenarios. The programme maps the cornering speed to the corresponding turn radius for each track segment with a turn radius below 50 metres; for segments with a turn radius exceeding this threshold, the vehicle's maximum speed is assigned directly. The BSP is thereby obtained by mapping the cornering speed values with track curvatures.

The simulation then proceeds to determine the vehicle's maximum achievable longitudinal acceleration at each track position. Since the G-G-V diagram encompasses both acceleration and braking regimes, a given combination of speed and lateral acceleration yields two distinct values of A_x. The G-G-V diagram is therefore partitioned into two separate regions to enable independent extraction of acceleration and braking data.

Two sequential iteration loops are implemented to update the speed values derived from the Boundary Speed Profile. The first loop evaluates the maximum achievable longitudinal acceleration at each instance using the acceleration Performance Envelope, while the second loop applies the corresponding braking deceleration. The complete Limit Speed Profile is thereby obtained as the final output of the dynamic event simulation.

In addition to speed values, more vehicle parameters can be simulated to provide further insights to the team.

5.3.3 Subsystem Simulation Implementation

5.3.3.1 Aerodynamics

Aerodynamic properties are provided by the Aerodynamics department based on Computational Fluid Dynamics (CFD) simulations conducted using a third-party solver. Simulations are performed across a range of flow directions and vehicle geometries to account for both straight-line and cornering operating conditions, such that the aerodynamic coefficients vary accordingly between these regimes. While it is desirable to further differentiate the aerodynamic properties as a function of cornering radius, any such differentiation must be implemented in a manner that preserves the linearity and differentiability of the optimisation objective function. To satisfy this requirement, a sigmoid function is employed in place of conventional if-else conditional logic.

\[\sigma = \frac{1}{1 + e^{-k\left(\sqrt{\delta_{in}^2 + \varepsilon} - \frac{2\pi}{180}\right)}} \tag{5.22}\] \[CL = CL_s(1 - \sigma) + CL_c \cdot \sigma \tag{5.23}\] \[CD = CD_s(1 - \sigma) + CD_c \cdot \sigma \tag{5.24}\] \[ab = ab_s(1 - \sigma) + ab_c \cdot \sigma \tag{5.25}\]

5.3.3.2 Steering System

Ackermann steering geometry is widely adopted to differentiate the steering angles of the inner and outer front wheels during cornering, thereby improving directional stability and reducing tyre scrub. The simulation of Ackermann geometry effects on vehicle dynamic performance is therefore necessary for accurate modelling.

The Ackermann geometry is characterised using Optimum Kinematic software, which generates the relationship between the left and right steering angles and the percentage of maximum steering input as a function of the expected cornering radius. Polynomial curve fitting is subsequently applied to this data to establish a continuous functional relationship between the inner and outer wheel steering angles and the instantaneous cornering radius. Incorporating this relationship into the slip angle formulation modifies the front axle slip angle expressions as follows:

\[\alpha_{FR} = -\delta_{\text{Inner}} + \arctan\!\left(\frac{V_y + \omega_r \cdot a}{V_x + \omega_r \cdot d/2}\right) \tag{5.26}\] \[\alpha_{FL} = -\delta_{\text{Outer}} + \arctan\!\left(\frac{V_y + \omega_r \cdot a}{V_x - \omega_r \cdot d/2}\right) \tag{5.27}\] \[\alpha_{RR} = \arctan\!\left(\frac{V_y - \omega_r \cdot a}{V_x + \omega_r \cdot d/2}\right) \tag{5.28}\] \[\alpha_{RL} = \arctan\!\left(\frac{V_y - \omega_r \cdot a}{V_x - \omega_r \cdot d/2}\right) \tag{5.29}\]

This now enables the comparison between different Ackermann geometry as shown below.

Simulation results comparing the vehicle performance with and without the Ackermann steering correction were evaluated. Upon overlaying the two outputs, the resulting G-G diagrams were found to be almost entirely coincident, with differences manifesting only at the level of decimal precision.

It was further observed that the introduction of the Ackermann steering feature introduces additional numerical complexity into the optimisation problem, arising from the higher-order polynomial curve fitting function used to represent the steering geometry.

Given that the performance benefit of the Ackermann steering feature is marginal while its adverse effect on numerical stability is significant, it was decided that this feature would be retained exclusively for steady-state G-G diagram simulation and deliberately excluded from the quasi-steady-state lap simulation.

5.3.3.3 Brake System

During braking, longitudinal load transfer shifts the normal load distribution towards the front axle. To fully exploit the available tyre grip under these conditions, it is advantageous to distribute the braking force asymmetrically between the front and rear axles, commonly referred to as brake bias. Determining the optimal brake bias for the NUS FSAE vehicle under its specific operating conditions is therefore of direct engineering interest, and a dedicated simulation capability was developed to address this requirement.

The brake bias constraint is implemented by imposing an additional equality constraint within the numerical optimisation framework, requiring that the ratio of front to total braking force is equal to the prescribed brake bias value during braking.

\[\text{Brake Bias} = \frac{F_{xFL}}{F_{xFL} + F_{xRL}} \tag{5.30}\]

By adding this constraint, it is observed from the G-G diagram comparison that the introduction of a fixed brake bias reduces the peak longitudinal deceleration relative to the unbiased baseline. As lateral acceleration increases, aerodynamic and inertial load transfer shifts the tyre normal forces away from the balance point assumed by the fixed bias, causing one axle to saturate before the other reaches its grip limit. The combined braking capacity is therefore lower than the theoretical maximum achievable when both axles are allowed to operate simultaneously at their respective traction limits.

More critically, the fixed brake bias introduces a discrete regime switch into the simulation model, causing the objective landscape presented to the optimiser to contain a discontinuity at which the gradient of the braking force with respect to the decision variables is undefined. This manifests visibly in the G-G diagram as the inward concavity observed in the high-lateral-acceleration region of the constrained curve, where the optimiser loses the ability to track the outer performance envelope smoothly. The underlying cause of this instability and its impact on the fidelity of the simulation results remains an open issue.

Consequently, the brake bias constraint is deliberately excluded from the quasi-steady-state (QSS) lap simulation in order to preserve its numerical stability and the physical continuity of the resulting speed and force profiles.

6.2.1 Numerical Optimization Tuning

At each point of interest within the simulation, a separate numerical optimisation is performed to determine the optimal set of decision variables. Since a generalised configuration of numerical constraints and initial guesses is applied uniformly across all instances, certain optimisation problems may fail to converge due to local infeasibility or the highly nonlinear nature of the objective landscape in certain regions of the operating domain. To prevent isolated failures from interrupting the overall simulation process, the programme skips any failed instances automatically and continues execution. Upon completion of the simulation, an optimisation success rate is computed as the ratio of successfully converged instances to the total number of optimisation attempts, providing a quantitative metric for assessing the overall credibility and reliability of each simulation run.

Several additional design measures were implemented to improve the optimisation success rate. At each query point, the initial guesses for the decision variables are set to the optimised values obtained from the immediately preceding query point, rather than from fixed arbitrary values. Additional measures include the adaptive modification of numerical constraint bounds for certain decision variables within higher speed regimes, where the physical operating conditions deviate significantly from those at lower speeds.

6.2.2 Autocross and Endurance Simulation Validation

The simulation results were validated by comparison against telemetry data recorded from the NUS FSAE vehicle of the preceding season, designated R25E, during the Endurance event at the 2025 Formula SAE Electric Competition. To ensure a fair and consistent comparison, the LTS26 simulation was conducted under conditions identical to those of R25E, and the results were additionally compared against the outputs of the preceding LTS25 model.

The simulated speed profile was found to be in close agreement with the competition telemetry data, both in terms of the overall driving pattern and the peak cornering speeds achieved along the track.

Discrepancies between the simulated and recorded speed profiles were observed primarily in the high-speed regions of the run, where the vehicle experienced significant powertrain thermal management constraints resulting in substantial losses in energy transmission efficiency. As the current simulation models the powertrain based on its theoretical torque-speed characteristics without accounting for thermal dissipation effects, it is unable to replicate the performance degradation that occurs under sustained high-load operating conditions in competition.

Furthermore, the simulation was unable to fully replicate the dynamic behaviour of a human driver on track sections such as slaloms, where the turn radius varies rapidly over short distances. This discrepancy reflects an inherent limitation of the quasi-steady-state modelling approach itself, which assumes the vehicle is in equilibrium at each instant and is therefore unable to capture the transient dynamics. Resolving this limitation will ultimately require a transition to a fully transient dynamic simulation framework, which is identified as a long-term development objective in Section 7.

Despite these limitations, LTS26 demonstrated a substantial improvement in computational efficiency relative to its predecessor. As a four-wheel quasi-steady-state model, LTS26 required approximately half the computation time of LTS25, which operated on a simpler two-wheel bicycle model. This improvement not only satisfies the computational efficiency objective established at the outset of the development programme, but also demonstrates that the more complex four-wheel model can be evaluated more efficiently through the decoupled Performance Envelope architecture introduced in this work. This result further highlights the potential for continued expansion of the vehicle model towards greater fidelity and complexity in future development cycles.