Lap time and delta analysis

Lap time analysis decomposes a lap into distance-aligned events and state variables (speed, longitudinal/lateral acceleration, throttle/brake, gear) to attribute time gains and losses to specific driver inputs, tyre states, aerodynamic conditions, and traffic effects. The goal is a quantitative, reproducible explanation of delta-time traces, enabling strategy and setup decisions (brake bias, aero level, tyre plan, shift maps).

Data channels and sampling

Recommended minimum channels (100–500 Hz acquisition, 50–100 Hz processing):

• Vehicle speed (km/h) and distance (m)
• Longitudinal / lateral acceleration (g)
• Brake pressure (bar), throttle (%), steering angle (°)
• Gear, engine speed (rpm), rear-axle power estimate (kW)
• GPS position (lat/lon) → projected centreline abscissa s (m)
• Tyre inner/middle/outer IR (°C) where permitted
• Ride heights or aero platform proxy (if available)

Distance alignment and rolling delta

Analysis must compare laps on a common distance axis (not time) to preserve causality through braking/accel phases.

Curvilinear abscissa (distance) $s(t)\;=\;\int _{0}^{t}v(\tau )\,\mathrm {d} \tau \quad \Rightarrow \quad x(s){\text{ is any channel resampled on }}s\in [0,\;L_{\mathrm {lap} }].$

Rolling delta (cumulative time difference) For two laps A (reference) and B (candidate), integrate the local time step over distance: $\Delta T(s)\;=\;\int _{0}^{s}\left({\frac {1}{v_{B}(\sigma )}}-{\frac {1}{v_{A}(\sigma )}}\right)\,\mathrm {d} \sigma \cdot \Delta s.$

This yields the familiar continuous “time-gain/time-loss” trace engineers use to pinpoint where a lap diverges (brake point, minimum-speed, exit). A practical workflow is: (i) compute s from GPS/speed, (ii) resample all channels on s (e.g., 1 m), (iii) form $\Delta T(s)$, (iv) annotate corner entries/apices/exits, (v) attribute deltas.

Corner phase metrics

Define corner regions by decel/steer thresholds and apex index $s_{\mathrm {apex} }$. Report phase metrics lap-to-lap:

Metric	Definition (distance domain)	Typical F1 range
Entry speed	$v(s_{\mathrm {entry} })$ at first brake-on	220–320 km/h
Minimum speed	$\min _{s\in {\text{corner}}}v(s)$	60–180 km/h
Brake zone length	$s_{\mathrm {brake\,off} }-s_{\mathrm {brake\,on} }$	80–160 m
Exit delta @150 m	$\Delta T(s_{\mathrm {apex} }+150\,\mathrm {m} )$	±0.05–0.25 s
Lateral peak	$\max a_{y}$ (filtered)	3.5–5.5 g

Physics back-bone (friction ellipse & power)

Friction-ellipse constraint (per tyre) maps combined usage: $\left({\frac {F_{x}}{\mu F_{z}}}\right)^{2}+\left({\frac {F_{y}}{\mu F_{z}}}\right)^{2}\;\leq \;1,$ so braking deep (high $F_{x}$) reduces available lateral $F_{y}$ and sets the attainable entry/rotation trade-off. Longitudinal acceleration is power-limited at high speed: $a_{x}\approx {\frac {P_{\mathrm {drv} }}{m\,v}}\;-\;{\frac {D(v)}{m}},\qquad D(v)={\tfrac {1}{2}}\rho C_{D}Av^{2},$ governing straight-line delta growth when one lap has higher deployment or lower drag.

Fuel and mass sensitivity

A first-order lap-time penalty with fuel mass: $\Delta t_{\mathrm {fuel} }(m)=k_{f}\,m,\qquad k_{f}\approx 0.030{\text{–}}0.040~\mathrm {s\,kg^{-1}\,lap^{-1}} {\text{ (2022+ era)}}.$ Use per-sector sensitivity when fuel burn is uneven (e.g., long WOT sectors).

Tyre degradation imprint on delta

Represent compound-specific degradation as convex in age a (laps since stop): $\Delta t_{\mathrm {deg} }(c,a)\;=\;\alpha _{c}\,a\;+\;\beta _{c}\,a^{2},$ calibrated from long-run pace. A typical (illustrative) prior:

Compound	\alpha_c (s/lap)	\beta_c (s/lap²)	Nominal stint (laps)
C1	0.015	0.00018	25–35
C3	0.025	0.00035	16–24
C5	0.035	0.00070	8–15

Tie this to carcass/bulk temperatures via a window penalty if you track IR channels.

Sector attribution (worked method)

1) Compute $\Delta T(s)$ and mark sector endpoints $s_{k}$.

2) Sector deltas are differences of the cumulative curve: $\Delta T_{k}=\Delta T(s_{k})-\Delta T(s_{k-1})$.

3) Within a corner, split phase deltas by integrating only over the corresponding s intervals (entry / mid / exit).

4) Attribute mechanisms by co-evaluating $v,\,a_{x},\,a_{y},\,\mathrm {brake} ,\,\mathrm {throttle}$ traces and, where available, power and platform proxies.

Example comparison (same car, two laps)

Item	Lap A (ref)	Lap B	Comment
Tlap (s)	88.420	88.205	B faster −0.215 s
S1 delta (s)	–	−0.090	Later brake, same Vmin
S2 delta (s)	–	−0.055	Higher exit accel (deployment)
S3 delta (s)	–	−0.070	Lower drag in final straight
Min speed T9 (km/h)	146	144	B slower at apex, but better exit
Exit delta @150 m (s)	–	−0.060	Time gained after apex

Traffic/dirty air correction (optional)

When comparing laps with different traffic states, incorporate a penalty term: $\Delta t_{\mathrm {traffic} }(s)=\lambda \,u(s)\;-\;\eta \,z(s),$ where $u(s)=1$ inside a user-defined “close-following” gap (dirty-air zone), and $z(s)=1$ when DRS is active. Calibrate $\lambda ,\eta$ from multi-lap data (typ. 0.15–0.60 s/lap impact in prolonged following; DRS gain 0.10–0.30 s/lap depending on zones).

Statistical modelling & validation

• Back-to-back deltas: same stint, same traffic → isolates driver inputs.
• Regression on distance grid: fit $\Delta T(s)$ on covariates $[v,a_{x},a_{y},\mathrm {brake} ,\mathrm {throttle} ]$ to quantify marginal effects.
• Optimum-lap synthesis: compare observed $v(s)$ to minimum-time solution (QSS/optimal control) to reveal theoretical headroom.
• Tooling: most pro suites provide rolling best/“theoretical best” from micro-sectors using distance alignment.

Minimum-time benchmarks (for context)

Quasi-steady-state (QSS) and optimal-control solvers produce reference speed/acceleration profiles under tyre-load and power limits. These are invaluable to test whether an observed delta stems from sub-optimal inputs or hard constraints (power, drag, μ). See surveys and theses for reproducible formulations and open data.

Practical checklist

• Align on distance, not time.
• Inspect $v(s)$, brake, throttle, $a_{x},a_{y}$ around every apex.
• Quantify entry (brake point & decel), rotation (Vmin), exit (accel to +150 m).
• Separate fuel, tyre age, deployment state, and traffic before blaming driver.
• Validate conclusions against a minimum-time or “theoretical best” lap.

References

• FIA Regulations Hub
• AiM RaceStudio 3 Manual – rolling/theoretical lap features
• Dal Bianco et al. (2017): Minimum-time optimal control simulation of a GP2 race car – Univ. of Southampton
• Veneri (2016): Minimum Lap Time of Race Cars – Univ. of Padova
• TU/e (2023): Optimisation methods for QSS lap-time simulation – Eindhoven
• Borsboom et al. (2020): Convex optimisation framework for minimum lap time – TU/e / IEEE TVT
• Reinero (2021): QSS lap-time simulator for single-seater EV – PoliTo
• Oregon State (2018): Trajectory-optimised lap-time simulation – thesis
• Bonfim (2023): Transient model & NMPC for lap-time – USP
• “Getting Faster, One Second at a Time” – delta-t primer
• Intro to delta-t and distance alignment – article