Editing 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)'''
<math>
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}}].
</math>

'''Rolling delta (cumulative time difference)'''
For two laps A (reference) and B (candidate), integrate the local time step over distance:
<math>
\Delta T(s)
\;=\;
\int_{0}^{s}
\left(\frac{1}{v_B(\sigma)} - \frac{1}{v_A(\sigma)}\right)\,\mathrm{d}\sigma \cdot \Delta s.
</math>

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 <math>\Delta T(s)</math>, (iv) annotate corner entries/apices/exits, (v) attribute deltas.

== Corner phase metrics ==

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

{| class="wikitable"
! Metric !! Definition (distance domain) !! Typical F1 range
|-
| Entry speed || <math>v(s_{\mathrm{entry}})</math> at first brake-on || 220–320 km/h
|-
| Minimum speed || <math>\min_{s \in \text{corner}} v(s)</math> || 60–180 km/h
|-
| Brake zone length || <math>s_{\mathrm{brake\,off}}-s_{\mathrm{brake\,on}}</math> || 80–160 m
|-
| Exit delta @150 m || <math>\Delta T(s_{\mathrm{apex}}+150\,\mathrm{m})</math> || ±0.05–0.25 s
|-
| Lateral peak || <math>\max a_y</math> (filtered) || 3.5–5.5 g
|}

== Physics back-bone (friction ellipse & power) ==

Friction-ellipse constraint (per tyre) maps combined usage:
<math>
\left(\frac{F_{x}}{\mu F_z}\right)^2 + \left(\frac{F_{y}}{\mu F_z}\right)^2 \;\le\; 1,
</math>
so braking deep (high <math>F_x</math>) reduces available lateral <math>F_y</math> and sets the attainable entry/rotation trade-off. Longitudinal acceleration is power-limited at high speed:
<math>
a_x \approx \frac{P_{\mathrm{drv}}}{m \, v} \;-\; \frac{D(v)}{m},
\qquad
D(v)=\tfrac{1}{2}\rho C_D A v^2,
</math>
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:
<math>
\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)}.
</math>
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):
<math>
\Delta t_{\mathrm{deg}}(c,a) \;=\; \alpha_c\, a \;+\; \beta_c\, a^2,
</math>
calibrated from long-run pace. A typical (illustrative) prior:

{| class="wikitable"
! 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 <math>\Delta T(s)</math> and mark sector endpoints <math>s_k</math>.  

2) Sector deltas are differences of the cumulative curve: <math>\Delta T_k=\Delta T(s_k)-\Delta T(s_{k-1})</math>.  

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

4) Attribute mechanisms by co-evaluating <math>v,\,a_x,\,a_y,\,\mathrm{brake},\,\mathrm{throttle}</math> traces and, where available, power and platform proxies.

'''Example comparison (same car, two laps)'''

{| class="wikitable"
! Item !! Lap A (ref) !! Lap B !! Comment
|-
| T<sub>lap</sub> (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:
<math>
\Delta t_{\mathrm{traffic}}(s)
= \lambda\, u(s) \;-\; \eta\, z(s),
</math>
where <math>u(s)=1</math> inside a user-defined “close-following” gap (dirty-air zone), and <math>z(s)=1</math> when DRS is active. Calibrate <math>\lambda,\eta</math> 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 <math>\Delta T(s)</math> on covariates <math>[v,a_x,a_y,\mathrm{brake},\mathrm{throttle}]</math> to quantify marginal effects.  
* Optimum-lap synthesis: compare observed <math>v(s)</math> 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 <math>v(s)</math>, brake, throttle, <math>a_x,a_y</math> 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 ==
* [https://www.fia.com/regulation/category/110 FIA Regulations Hub]
* [https://www.aim-sportline.com/docs/racestudio3/manual/latex/racestudio3-manual-en-latest.pdf AiM RaceStudio 3 Manual – rolling/theoretical lap features]
* [https://eprints.soton.ac.uk/417133/1/GP2manuscriptPURE_002_.pdf Dal Bianco et al. (2017): Minimum-time optimal control simulation of a GP2 race car – Univ. of Southampton]
* [https://thesis.unipd.it/retrieve/8c327453-5809-425a-b416-3cfad77b5f07/Veneri_Matteo_tesi.pdf Veneri (2016): Minimum Lap Time of Race Cars – Univ. of Padova]
* [https://research.tue.nl/files/318078060/1514032-Applications_of_Optimization_Methods_to_Quasi_Steady-State_Lap_Time_Simulation.pdf TU/e (2023): Optimisation methods for QSS lap-time simulation – Eindhoven]
* [https://pure.tue.nl/ws/files/174109985/Borsboom.Fahdzyana.ea.TVT20.pdf Borsboom et al. (2020): Convex optimisation framework for minimum lap time – TU/e / IEEE TVT]
* [https://webthesis.biblio.polito.it/20760/1/tesi.pdf Reinero (2021): QSS lap-time simulator for single-seater EV – PoliTo]
* [https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/ft848x05t Oregon State (2018): Trajectory-optimised lap-time simulation – thesis]
* [https://sites.usp.br/ldsv/wp-content/uploads/sites/1453/2024/07/Douglas-Bonfim-Transient_model_based_laptime_simulation_of_a_race_car-compressed.pdf Bonfim (2023): Transient model & NMPC for lap-time – USP]
* [https://www.nessoft.com/ispeed/one_second_at_a_time.pdf “Getting Faster, One Second at a Time” – delta-t primer]
* [https://www.yourdatadriven.com/introduction-to-motorsport-data-analysis-delta-t/ Intro to delta-t and distance alignment – article]