Cameron

Introduction — From Race Cars to Inverse Problems

In professional motorsport, drivetrain parameters such as gear ratios and torque curves are known quantities. They are defined during design, validated in simulation, and measured on dynos. When analysing gearbox loading or shaft stresses, engineers typically start with the internal parameters and check them against reality.

But what if you only have the outside of the system?

Publicly available Formula One telemetry provides vehicle speed, engine RPM, and gear selection, but not gear ratios, drivetrain efficiency, or engine torque. That gap turns a familiar forward calculation into something more interesting — an inverse problem.

Model Structure

Functional form

y(t) ≈ F(x(t), θ) + ε(t)

Definitions

y(t) — measured output signals (speed, RPM)
x(t) — known operating conditions (gear selection)
θ — unknown model parameters (gear ratios, resistive coefficients)
ε(t) — stochastic noise and modelling error

Instead of predicting speed from known ratios (a forward problem), we use speed to infer the ratios and then reconstruct the hidden quantity of interest, being the torque.

This pattern is not unique to motorsport. It is the foundation of system identification and inverse modelling across engineering and science [1,2]. The torque-from-telemetry exercise is simply a compact, accessible example of a much broader methodological class.

And that is where the real relevance lies.

The Journey — Building the Model Backwards

Rather than beginning with optimisation, the project began with physics.

1. From Speed to Torque — The Physical Backbone

If vehicle speed is known, acceleration can be estimated. From acceleration, Newton’s second law gives the force at the wheels. With aerodynamic drag and rolling resistance added, wheel torque follows from tyre radius. Propagating torque upstream through gear ratios and drivetrain efficiency yields gearbox and engine torque.

Conceptually, this is simple. In practice, two immediate problems emerged:

Acceleration requires numerical differentiation, and differentiation amplifies noise.
Aerodynamic and rolling resistance models are simplifications; real systems violate them.

Differentiation is well known to be an ill-conditioned operation, where in this case, small perturbations in speed can produce large fluctuations in acceleration estimates [3], a structural feature of inverse problems. Smoothing and regularised differentiation techniques exist precisely because naïve finite differences destabilise inference [3].

At this stage, it became clear that the difficulty was managing noise and model mismatch.

2. Identifying the Drivetrain — Optimisation Under Imperfect Data

The next step was to infer:

Final drive ratio
Eight individual gear ratios

The idea was straightforward: engine RPM, divided by gear ratio and final drive, should match measured vehicle speed. The optimisation objective was to minimise the mean squared error between predicted and observed speed.

SciPy’s derivative-free Powell method was used, as it is often considered preferable when models are piecewise (due to gear selection) and gradients are awkward to compute [4].

Initial results were unstable. The optimiser found parameter combinations that reduced error even if that meant compensating for unmodelled physics. When the model was incomplete, optimisation “explains away” discrepancies by distorting parameters., showing that the limiting factor was not the optimiser, it was the data.

3. Cleaning the Data — Handling Regimes the Model Cannot Explain

Gear shifts introduce transient dynamics: clutch disengagement, torque cuts, and rapid RPM variation. The steady-state kinematic model linking RPM to speed simply does not apply in these windows. If those segments were included, the optimiser warped gear ratios to accommodate behaviour the model could never represent.

The solution was explicit regime handling:

Detect gear changes
Exclude short windows around each shift
Restrict optimisation to quasi-steady segments

Once shift transients were removed, optimisation converged cleanly to physically reasonable ratios. This showed that for inverse problems, parameter credibility can depend more on data curation and model adequacy than on algorithm sophistication.

4. Reconstructing Internal Torque

With gear ratios identified and acceleration stabilised via smoothing, the model could be run “forward”:

Measured speed → smoothed acceleration
Acceleration → wheel force
Wheel force → wheel torque
Wheel torque → gearbox output torque
Gear ratios + efficiency → engine torque

This produced per-gear torque traces and peak loading estimates, effectively reconstructing internal drivetrain loads from external measurements alone. The torque curves were not perfect, reflecting modelling assumptions (constant drag coefficient, fixed efficiency), but they achieved coherence and plausibility.

Why This Matters Beyond Motorsport

In a controlled gearbox development programme, engineers already know the ratios and torque map. So although this project is not pertinent to that specific industry, it is to settings where internal quantities are inaccessible.

Inverse inference becomes essential when:

Direct sensors are infeasible, intrusive, or too expensive
Parameters drift due to ageing or manufacturing variability
Systems operate in the field and cannot be dismantled
Only operational data is available

This pattern appears across domains:

Aerospace structures: reconstructing loads from strain measurements [5]
Wind turbine drivetrains: identifying torsional loads from sparse sensors
Battery management systems: estimating internal electrochemical state from voltage/current via filtering methods [6]
Thermal engineering: inferring surface heat flux from internal temperature measurements (inverse heat conduction) [7]
Medical imaging and tomography: reconstructing internal structure from indirect projections

In each case, the forward model is known (or partially known), but the quantities of interest are hidden. Optimisation or filtering bridges that gap, as these problems are often ill-posed: small measurement errors can produce large parameter variations unless regularisation, smoothing, or constraints are applied [3].

What this project demonstrates, in miniature, is the full lifecycle of such a workflow:

Construct a physically grounded forward model
Formulate an optimisation problem to infer parameters
Identify and remove regimes the model cannot represent
Stabilise noise-amplifying operations (differentiation)
Validate inferred quantities for physical plausibility

Where Machine Learning Fits

Machine learning is often discussed as a replacement for physics-based modelling. In practice, its strongest role is frequently supportive:

Denoising signals prior to inversion
Learning surrogate forward models
Correcting systematic residual errors
Quantifying uncertainty

In many operational systems, including battery management and weather forecasting, learned components sit inside a fundamentally physics-based estimation framework [2,6].

Outcomes

Inferred drivetrain ratios from telemetry alone
Reconstructed wheel, gearbox, and engine torque traces
Identified the impact of transient regimes on parameter distortion
Demonstrated noise sensitivity in numerical differentiation
Built a complete optimisation-based inverse modelling pipeline in Python

More importantly, the project highlighted that inverse inference is rarely limited by mathematics — it is limited by noise, model adequacy, and data discipline.

The full implementation is available on my GitHub.

References

[1] Ljung, L. System Identification: Theory for the User.
[2] Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation.
[3] Engl, Hanke, Neubauer. Regularization of Inverse Problems.
[4] SciPy Documentation — Powell Method (Derivative-Free Optimisation).
[5] NASA Technical Reports on inverse strain-based load reconstruction for aerospace structures.
[6] Plett, G. L. Extended Kalman filtering for battery management systems, Journal of Power Sources.
[7] Beck, J. V. et al. Inverse Heat Conduction: Ill-Posed Problems.

Inferring Torque from Telemetry