A new preprint proposes a way to calculate feedforward steering inputs—precomputed steering commands—for a car with many trailers. The paper reports that the conventional symbolic route becomes intractable at about five trailers, a scaling problem the proposed method is designed to avoid.
A sharp split in the benchmark
The comparison focused on three implementations of one elementary derivative-computation block: direct symbolic differentiation with SymPy; Bell polynomials, which organise the many terms in Faà di Bruno composition; and the proposed truncated formal-power-series method, which keeps only the terms needed up to the requested derivative order. The algorithm combines angular intermediate variables, factorised single-trailer sub-maps, and rescaled product and quotient rules.
On the reported test, SymPy hit its 20-second timeout at derivative order 14. The Bell-polynomial route completed order 24 in 7.7 seconds, then exceeded its 60-second timeout by order 32. The paper’s prose reports cubic scaling for the proposed method and performance below one millisecond through order 40, although one table entry at that order conflicts with the prose claim.
Where the Bell-polynomial calculation completed, its relative error against the proposed method was about 10 to the minus 15, described as machine precision in IEEE 754 double arithmetic. The agreement is not an independent exact-ground-truth check across all orders, because the comparison covers only cases in which the Bell-polynomial method completed.
From derivatives to a parking manoeuvre
The full pipeline is reported to scale as the fourth power of the trailer count and to run well under one second for 20 trailers. The paper also states that it computed and animated a 20-trailer parking manoeuvre, providing an end-to-end illustration of the method.
What the benchmark cannot settle
Those timings came from a single core of a 2024 laptop running Python 3.12, SymPy 1.14 and NumPy 2.2; per-row timeouts were 20 seconds for direct symbolic differentiation and 60 seconds for the Bell-polynomial route. The resulting cutoffs are therefore benchmark- and implementation-dependent, rather than universal limits on trailer count.
The derivative test used one elementary form, h(s) = cos g(s), evaluated at a fixed point. For each derivative order r, g was a degree-r polynomial with integer coefficients drawn once using a fixed pseudorandom seed. Whether the reported performance generalises to other analytic functions, coefficient distributions or path profiles remains open.
The study offers only a qualitative parking animation; it reports no physical-vehicle testing, collision analysis, steering-constraint validation or tracking-error metrics. The time reparametrization used in the example is explicitly heuristic, so the study does not establish bounded steering velocity or acceleration.
The authors interpret the O(n^4) pipeline as expected to fit real-time budgets for up to 20 trailers. That remains an expectation from computation and simulation, not a demonstration of real-time operation under disturbances or modelling errors, and the study does not compare it with every symbolic, automatic-differentiation or numerical alternative.
Paper data and sources
Original title: Efficient flatness-based computation of trajectories for vehicles with many trailers
Authors: Basile Graf, Gustave Lapierre, Philippe Müllhaupt
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text