Disturbance-aware minimum-time planning strategies for motorsport vehicles with probabilistic safety certificates

Marco Gabiccini; Martino Gulisano; Massimo Guiggiani; Matteo Masoni

REVIEW 4 major objections 5 minor 28 references

Disturbance-aware minimum-time planning strategies for motorsport vehicles with probabilistic safety certificates

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Minimum-lap-time planning can be made disturbance-aware and probabilistically safe by propagating state covariance through the optimization, and by co-optimizing a time-varying LQR feedback law the closed-loop plan meets the safety…

desk verdict A competent and genuinely useful planning extension, but the 'probabilistic safety certificates' in the title amount to a first-order approximation with no quantified nonlinearity error and no open-loop validation. read the letter →

arxiv 2506.13622 v1 pith:GFDGTBLF submitted 2025-06-16 cs.RO

Martino Gulisano , Matteo Masoni , Marco Gabiccini , Massimo Guiggiani This is my paper · ORCID

classification cs.RO

keywords minimumlap-timetrajectoryplanningstochasticvehicledynamicsprobabilisticsafetycertificateschanceconstraintstime-varyingLQRcovariancepropagationrobustoptimalcontrolautonomousracing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that minimum-lap-time trajectory planning can be made robust to disturbances without sacrificing speed, by embedding uncertainty propagation and feedback action directly in the planning stage. Two planners are developed: an open-loop one that propagates the state covariance over a fixed H-step horizon and uses the worst-case growth to tighten track and tire-friction constraints, and a closed-loop one that co-optimizes a time-varying LQR feedback law with the trajectory and propagates the covariance through the controlled dynamics. On a representative Barcelona-Catalunya sector, both schemes meet the prescribed safety probability, the nominal non-robust trajectory is infeasible under the same disturbances, and the closed-loop variant achieves this with smaller lap-time penalties than the open-loop worst-case method. A sympathetic reading takes the main contribution to be a concrete recipe for converting chance constraints into deterministic back-offs that remain tight enough to keep the car fast.

What carries the argument

The central object is the state covariance matrix $P(t)$ propagated along the mean trajectory by the Lyapunov equation, together with the back-off construction $\beta_i = \gamma\sigma_i$ that converts the chance constraint $\Pr\{h_i(x)\le 0\}\ge p$ into the deterministic constraint $h_i(\mu,u)+\beta_i\le0$. In the closed-loop variant the same object is propagated through the controlled dynamics $\tilde{A}(t)=A(t)-B(t)K(t)$, with the time-varying LQR gain $K$ appearing as an optimization variable; this is what lets the planner estimate how much the driver's feedback will attenuate disturbances before it decides how much to tighten the constraints.

What would settle it

Run many noise realizations (e.g., 10,000) of the full nonlinear single-track model following the closed-loop robust plan with its optimized LQR gain, counting track-limit and friction-ellipse violations; if the empirical violation frequency exceeds the prescribed p (e.g., 10% at gamma=1.28) beyond sampling error, the certificate is falsified.

Watch

Extended reading notes

Core claim

The central claim is that probabilistic safety certificates can be embedded directly in minimum-time trajectory optimization using first-order covariance propagation, and that modelling the driver's feedback action during planning makes the resulting trajectories both safer and faster than nominal planning or worst-case open-loop back-off. The mean state follows the deterministic single-track vehicle dynamics, while the covariance evolves along the mean by the Lyapunov equation $\dot{P}(t) = A(t)P(t)+P(t)A(t)^T+Q(t)$; every inequality constraint is then tightened by a back-off $\beta_i = \gamma\sigma_i$, where $\gamma = \Phi^{-1}(p)$ and $\sigma_i$ is the standard deviation of the constraint's linearized gradient. The open-loop scheme resets the covariance to $P_0$ at every node and enforces constraints with the H-step-evolved worst-case covariance; the closed-loop scheme treats the feedback gain as a decision variable, propagates covariance through the controlled linearization $A(t)-B(t)K(t)$, and reports lower sector-time penalties (for example 1.39% versus 1.66% for track-limit back-off alone, and 2.07% versus 2.57% with both constraints). Noise-realization simulations with 100 random runs show the robustified LQR controller violating track limits less often than one tracking the nominal trajectory.

Load-bearing premise

The safety certificate rests on a linear approximation of the vehicle's nonlinear dynamics: if the true state distribution strays from the assumed Gaussian shape, the real violation rate can differ from the prescribed p, and no bound on that gap is given; the open-loop variant also forgets disturbances older than its H-step window by resetting the covariance.

Editorial extensions

If this is right

A nominal trajectory planned without uncertainty is infeasible under the tested noise level, so some disturbance-aware tightening is necessary, not optional, for safe minimum-lap-time references.
The closed-loop plan meets the same prescribed safety probability as the open-loop plan while paying smaller lap-time penalties, so modelling feedback in planning is a concrete way to reduce conservativeness.
The open-loop H-step scheme is a valid fallback when no trustworthy driver or controller model exists, provided H is chosen appropriately to bound unrealistically large covariance growth.
Back-off terms have a geometric meaning: they widen the vehicle footprint by a nonuniform margin that must stay inside the track, which gives a quantitative interpretation of safety margins in racing.
The resulting reference trajectories can be tracked by an LQR controller in autonomous driving, and for human drivers the safety fidelity increases with how well the time-varying LQR policy approximates the driver's behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is full-lap planning: because the closed-loop advantage comes from taming covariance growth, the gap between open- and closed-loop lap-time penalties should widen on longer horizons, which could be tested by solving the full circuit.
The open-loop covariance reset at each node suggests an online receding-horizon implementation: a planner that replans every H steps would obtain a feedback effect without explicitly modeling a controller, which may be a cheaper alternative for real-time deployment.
The same back-off logic could be applied to other performance-optimal trajectory problems with nonlinear dynamics and safety constraints, such as autonomous rally or urban racing, where heuristic margins are currently the norm.
If the linear-Gaussian approximation is the main limitation, a validation campaign at higher noise amplitudes or on low-friction conditions would show where the certificate starts to fail, informing whether a nonlinear covariance update is needed.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid, competent extension of chance-constrained and tube-based planning ideas to minimum-lap-time racing, and the closed-loop formulation—co-optimizing time-varying LQR gains inside the planner—is genuinely new relative to the cited MPC literature. The main problem is that the title and abstract call the result a 'probabilistic safety certificate,' but the guarantee is only a first-order Gaussian approximation, and the open-loop scheme is never validated. These are fixable overclaims.

What's new and good: The H-step covariance-reset open-loop scheme is a reasonable way to bound unbounded covariance growth without feedback, and the three-step closed-loop scheme (nominal plan, LQR synthesis, robustified replan with K as decision variables inside a trust region) is a natural and useful idea. The transcription to direct collocation with Lyapunov covariance propagation is standard but cleanly done. The sensitivity studies of gamma_TLC and P0 are helpful, and the comparison of open vs closed loop shows the expected trade-off: closed loop gives smaller lap-time penalties. The handling of mutually exclusive throttle/brake via smoothed indicators is a nice practical touch. Solver times are reported.

Soft spots: The 'certificate' language is not backed by a formal guarantee. Eq. (7f) uses beta = gamma*sigma from linearized Gaussian propagation; for the nonlinear single-track model with Pacejka tires and nonlinear friction/track constraints, the true violation probability can differ from p, and no bound is given. Worse, the open-loop scheme resets covariance to P0 at every node, so disturbances older than H steps are discarded; the safety claim is only local. And Sec. 5.3 explicitly says the open-loop analysis is not applicable because no stabilizing controller is computed, so the open-loop scheme is never simulated. The Monte Carlo validation covers only the closed-loop case, reports no violation counts or confidence intervals, and uses the same LQR structure and vehicle model—so it is not independent. The abstract's statement that 'both schemes meet the prescribed safety probability' is therefore not supported by the evidence presented.

Who this is for: researchers in autonomous racing and minimum-lap-time planning who want a practical recipe for adding disturbance robustness. They'll get useful ideas and a clear comparison. The theory won't give them a certified guarantee, but the numerical approach is reasonable.

Recommendation: Worth sending to peer review at a good robotics or vehicle-dynamics journal, with revisions. The authors should reword the safety certificate claims, disclose Q and the LQR weights, report empirical violation rates with confidence intervals, and either simulate the open-loop method or explicitly state that its guarantee is local over the H-step horizon. If they do that, the paper would be a solid contribution.

Referee Report

4 major / 5 minor

Summary. This manuscript addresses minimum-lap-time trajectory planning for a single-track vehicle subject to additive Gaussian disturbances. Two planners are formulated: an open-loop scheme that propagates a state covariance over a finite horizon H and tightens friction-ellipse and track-limit constraints with back-off terms, and a closed-loop scheme that embeds a time-varying LQR feedback policy and its associated Lyapunov covariance evolution directly into the optimal control problem. Both formulations are tested on a Barcelona-Catalunya sector, and the closed-loop planner is compared against a nominal trajectory in a Monte Carlo simulation with 100 noise realizations. The paper's central claim is that both schemes meet a prescribed probability of constraint satisfaction while the closed-loop variant incurs smaller lap-time penalties.

Significance. If substantiated, the contribution is practically valuable: it would demonstrate a tractable way to embed disturbance propagation and feedback attenuation into vehicle trajectory optimization, with an open-loop variant available when a feedback policy is not specified. The paper ships reproducible computational experiments (CasADi/IPOPT), reports solver timings and parameter sensitivity, and provides a clear comparison of two planning philosophies. However, the strength of the claimed 'probabilistic safety certificates' is not matched by the analysis: the certificates rest on a first-order Gaussian approximation with no quantified error bound, and the Monte Carlo validation is qualitative, reporting no violation counts, confidence intervals, or open-loop simulations. The methods are better described as heuristic robust planning tools rather than certified probabilistic guarantees, until the approximation error is bounded or empirically quantified.

major comments (4)

[Sec. 2.2, Eq. (7f)] The back-off term beta_i = gamma*sigma_i is derived by linearizing each constraint h_i around the mean and treating the linearized random variable as Gaussian; this is exact only when the dynamics are linear and the constraints are affine. Here the dynamics include the nonlinear Pacejka tire model (Sec. 2.1) and the constraints are nonlinear (friction ellipse, Eqs. (9)-(10), and the track-limit normal-projection constraint, Eqs. (11)-(12)), so the actual violation probability Pr{h_i(x) <= 0} can differ from the prescribed p. Since the title and abstract use the word 'certificate', the missing bound on the linearization error is load-bearing; either derive a bound or replace 'certificate' with a more modest claim and report empirical violation frequencies.
[Sec. 3, Eq. (13e) and Sec. 5.3] The open-loop planner resets the covariance to P0 at every grid node, so disturbances older than H steps are discarded and the constraint back-off at node k reflects only uncertainty within the last H steps. The safety claim for the open-loop scheme is therefore local to the H-step window, not a property of the full trajectory. Moreover, the paper does not simulate the open-loop trajectory at all: Section 5.3 explicitly states that this analysis is not applicable to the open-loop approach. Consequently, the abstract's statement that 'both schemes meet the prescribed safety probability' is not supported by any numerical evidence for the open-loop scheme.
[Sec. 5.3] The empirical validation reports 100 noise realizations for the closed-loop planner but provides no violation counts, no empirical probability estimates, and no confidence intervals; Fig. 11 is qualitative. The wording 'significantly fewer track violations' in the Conclusions (Sec. 6) is not quantified. Because the certificate claim hinges on matching a prescribed probability, the paper should report the number and rate of constraint violations (track-limit and friction-ellipse) in the Monte Carlo runs, with confidence intervals, and ideally for several values of p.
[Sec. 4.3, Eq. (17h)] The closed-loop formulation optimizes the feedback gains K_k jointly with the trajectory but constrains them to a trust region delta_K = 0.1*K_hat_k around the LQR gains computed for the nominal trajectory. No stability or performance guarantee is given for the final K_k along the re-planned trajectory; the Lyapunov covariance propagation in (17d) uses the decision-variable gains, but this is a consistency condition, not a certificate. Also, the smoothing parameter chi in Eq. (20) changes the effective complementarity constraint and therefore the meaning of the planned control; its influence on safety and lap time is not reported.

minor comments (5)

[Sec. 2.2] The phrase 'the higher the gain gamma1' should read 'the higher the gain gamma'; the subscript appears to be a typo.
[Fig. 5] The labels 'P0=0:257P0' and 'P0=47P0' should use proper fraction notation, e.g., 'P0 = (1/4) P0_bar' and 'P0 = 4 P0_bar'.
[Sec. 5.3] The text says '100 random initial conditions' for the closed-loop validation, while the caption of Fig. 11 says '100 simulations (each panel)'; clarify whether both panels use the same noise draws and whether the nominal-trajectory panel is included only as a reference.
[Sec. 2.3, Eq. (8g)] The algebraic variables z_k (contact forces) enter the back-off in Eq. (8g) but are treated as deterministic, whereas the state covariance is propagated; the paper should state explicitly that the algebraic variables are not randomized and discuss the consistency of this assumption.
[Sec. 2.2] The Lyapunov equation (4) and its solution (5) are introduced without a citation at the point of use; moving reference [24] to Eq. (4) would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the probabilistic back-off is derived from explicit Gaussian/linearization assumptions and user-set inputs, while the validation gaps for the open-loop scheme are evidentiary limitations rather than circular reductions.

full rationale

The paper's core derivation (Sec. 2.2, Eqs. (3)-(7)) starts from stated modeling assumptions: additive Gaussian noise, first-order linearization, Lyapunov covariance propagation, and quantile back-off beta_i = gamma sigma_i with gamma = Phi^-1(p). The safety probability p is an input, not a fitted output, and the covariance evolution follows from the assumed dynamics and noise covariance rather than from the desired safety conclusion. The closed-loop planner similarly derives covariance dynamics from the assumed LQR feedback law through A(t) - B(t)K(t), again from modeling structure rather than from the target result. No load-bearing self-citation appears: the Gabiccini self-citations (Refs. [7] and [8]) support only parallel minimum-lap-time solution background, not the safety-certificate claim. The genuine weaknesses are non-circular: (i) the Gaussian/first-order certificate has no quantified bound on nonlinearity error, so 'certificate' overstates the formality of the guarantee; (ii) Sec. 5.3 explicitly states 'This analysis is not applicable the open-loop approach', so the abstract's claim that 'both schemes meet the prescribed safety probability' is not empirically established for the open-loop scheme; and (iii) no Monte Carlo violation counts or confidence intervals are reported even for the closed-loop case. These are evidence and correctness concerns, not reductions of the derivation to its own inputs, and they do not meet the quoted-equation threshold for circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. Its load-bearing inputs are a hand-chosen initial covariance, an undisclosed noise covariance, undisclosed LQR weights, a hand-set prediction horizon, and the linearized Gaussian approximation that defines the safety certificate.

free parameters (6)

Initial covariance P0 = diag((0.1 m/s)^2, (0.01 m/s)^2, (0.01 rad/s)^2, (1 m)^2, (1 m)^2, (0.0175 rad)^2)
Chosen by hand based on typical range of variation of each state; directly sets the magnitude of the back-offs.
Open-loop prediction horizon H = 4
Critical design parameter that bounds how far covariance is propagated before reset; the paper notes it must be chosen to capture disturbances before driver response, but gives no selection criterion.
LQR weights W_k, R_k = Not reported
The time-varying LQR gains depend on these weights; they are never specified, so the closed-loop covariance, the back-offs, and the simulation results depend on undisclosed choices.
Noise covariance Q(t) = Not specified numerically
Assumed known and used in propagation and simulations, but no numerical values are given.
Gain trust-region deltaK = 0.1 * K_hat
Chosen to promote convergence; it limits how much the planned feedback can deviate from the nominal LQR.
Smoothing sharpness chi for indicator functions = Not specified
Controls the tanh approximation of the throttle/brake switching; affects gradients and the final control mapping.

assumptions (5)

domain assumption Disturbances are additive Gaussian white noise with known covariance Q(t).
Eq. (2) defines the perturbed dynamics; all subsequent covariance propagation and chance constraints rely on this model.
ad hoc to paper State distribution stays Gaussian and covariance follows the linearized Lyapunov equation.
First-order approximation in Sec. 2.2; not exact for the nonlinear vehicle model and provides no error bound.
ad hoc to paper Constraint violation probability is computed by linearizing each constraint around the mean and using the scalar quantile gamma = Phi^{-1}(p).
Eq. (7f) and back-off beta = gamma*sigma; exact only for linear constraints on a Gaussian state.
domain assumption A time-varying LQR controller is a good approximation of an expert driver's corrective behavior.
Stated in Sec. 4.2 and relied on for the closed-loop covariance propagation and the human-driving interpretation.
ad hoc to paper Integrating only the lower triangular part of P preserves symmetry and positive definiteness.
Empirical claim in Sec. 2.3; no proof is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Disturbance-aware minimum-time planning strategies for motorsport vehicles with probabilistic safety certificates." pith.science (2026). https://pith.science/paper/GFDGTBLF

@misc{pith2026250613622,
  author       = {Pith},
  title        = {Pith review of: Disturbance-aware minimum-time planning strategies for motorsport vehicles with probabilistic safety certificates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFDGTBLF}},
  note         = {Machine review of arXiv:2506.13622}
}

read the original abstract

This paper presents a disturbance-aware framework that embeds robustness into minimum-lap-time trajectory optimization for motorsport. Two formulations are introduced. (i) Open-loop, horizon-based covariance propagation uses worst-case uncertainty growth over a finite window to tighten tire-friction and track-limit constraints. (ii) Closed-loop, covariance-aware planning incorporates a time-varying LQR feedback law in the optimizer, providing a feedback-consistent estimate of disturbance attenuation and enabling sharper yet reliable constraint tightening. Both methods yield reference trajectories for human or artificial drivers: in autonomous applications the modelled controller can replicate the on-board implementation, while for human driving accuracy increases with the extent to which the driver can be approximated by the assumed time-varying LQR policy. Computational tests on a representative Barcelona-Catalunya sector show that both schemes meet the prescribed safety probability, yet the closed-loop variant incurs smaller lap-time penalties than the more conservative open-loop solution, while the nominal (non-robust) trajectory remains infeasible under the same uncertainties. By accounting for uncertainty growth and feedback action during planning, the proposed framework delivers trajectories that are both performance-optimal and probabilistically safe, advancing minimum-time optimization toward real-world deployment in high-performance motorsport and autonomous racing.

Figures

Figures reproduced from arXiv: 2506.13622 by the authors.

**Figure 1.** The lateral forces Y1 and Y2 are computed using a Pacejka’s Magic Formula, which depends on the total vertical load acting on the axle and the axle’s apparent slip angle. The longitudinal forces X1 and X2 are provided as inputs to the system. Specifically, the first two components of the input vector u correspond to acceleration and braking forces, while the third component represents the steering angle of the front… view at source ↗

**Figure 1.** Single Track model. state vector. Assuming a first-order approximation for the disturbance propagation rule, x(t) results in a Gaussian distribution with mean µ(t) and covariance P (t), so that x(t) ∼ N (µ(t), P (t)). Due to the symmetry of the probability density function (pdf) with respect to µ(t), it is possible to represent the time evolution of the pdf as: i) the deterministic evolution of the mean µ(t) µ˙(t) =… view at source ↗

**Figure 2.** Graphical representation of the two constraints analyzed in this section: the friction limit constraint in the XjYj plane (left panel), and the track limit constraint in the xy plane (right panel). In both figures, the filled gray area bounded by a dashed line indicates the accessible region in the presence of a back-off term. The points (Xj , Yj ) are constrained to lie within an ellipse whose semi-axes are given b… view at source ↗

Figures from the paper (9 more)

**Figure 3.** Figure 3: Schematic representation (depicted for H = 3) of the continuity and initialization equations associated with the H +1 propagated instances of the covariance matrix introduced in problem (13). The dashed rectangle highlights the k-th discretization step, at which the r…

**Figure 4.** Figure 4: The analyzed Catalunya circuit sector spans α ∈ [0.70, 0.77], with labeled checkpoints every ∆α = 0.01. It contains two corners – a low-speed turn (α = 0.72 − 0.73) and a high-speed turn (α = 0.75 − 0.76) – providing contrasting dynamic conditions for vehicle behavio…

**Figure 5.** Figure 5: Effect of the TLC penalty coefficient γ TLC (left panel) and the initial covariance P0 (right panel) on the optimal trajectory. Raising γ TLC from 0 to 3 increases the probability of satisfying the track-limit constraints from 50 % to 99 %. The right panel compares the…

**Figure 6.** Figure 6: Open-Loop method. Variations of longitudinal speed ∆u, steering angle ∆δ, longitudinal force ∆Fx, and lateral deviation ∆e relative to the nominal trajectory for three schemes — TLC (track-limit), FLC (friction-limit), and TLC+FLC (both). Dashed lines mark the sign of …

**Figure 7.** Figure 7: Open-Loop. Axle saturation Sj (x,u) for front axle (j=1, first panel) and rear axle (j=2, second panel) with respect to the curvilinear parameter α. Nominal configuration in dashed black, TLC in green, FLC in red, TLC+FLC in blue. solution, we now turn our attention to…

**Figure 8.** Figure 8: Closed-Loop method. Variations of longitudinal speed ∆u, steering angle ∆δ, longitudinal force ∆Fx, and lateral deviation ∆e relative to the nominal trajectory for three schemes — TLC (track-limit), FLC (friction-limit), and TLC+FLC (both). Dashed lines mark the sign o…

**Figure 9.** Figure 9: Closed-Loop. Axle saturation Sj (x,u) for front axle (j=1, first panel) and rear axle (j=2, second panel) with respect to the curvilinear parameter α. Nominal configuration in dashed black, TLC in green, FLC in red, TLC+FLC in blue. 19 [PITH_FULL_IMAGE:figures/full_fi…

**Figure 10.** Figure 10: Back-off envelopes for the open-loop (red) and closed-loop (cyan) robust planners. Each shaded band represents the area swept by the back-off-extended vehicle and must lie entirely within the actual track boundaries. The open-loop approach yields a wider envelope due …

**Figure 11.** Figure 11: 100 simulations (each panel) with random initial conditions and Gaussian noise realizations. In the left panel, the reference trajectory (black curve) is the nonrobust one, tracked using an LQR controller designed around it as per Step 2 of Subsection 4.2. In the rig…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

A free-trajectory quasi-steady-state optimal-control method for minimum lap-time of race vehicles

Veneri M, Massaro M. A free-trajectory quasi-steady-state optimal-control method for minimum lap-time of race vehicles. Vehicle System Dynamics. 2020 Jun;58(6):933–954

work page 2020
[2]

Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles

Lovato S, Massaro M. Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles. Vehicle System Dynamics. 2022 Nov;60(11):3650–3667

work page 2022
[3]

A three-dimensional free-trajectory quasi-steady-state optimal- control method for minimum-lap-time of race vehicles

Lovato S, Massaro M. A three-dimensional free-trajectory quasi-steady-state optimal- control method for minimum-lap-time of race vehicles. Vehicle System Dynamics. 2022 May;60(5):1512–1530

work page 2022
[4]

Comparison of direct and indirect methods for minimum lap time optimal control problems

Dal Bianco N, Bertolazzi E, Biral F, et al. Comparison of direct and indirect methods for minimum lap time optimal control problems. Vehicle System Dynamics. 2019 May; 57(5):665–696

work page 2019
[5]

A Direct/Indirect Approach to Optimal Control Problems

Bertolazzi E, Biral F. A Direct/Indirect Approach to Optimal Control Problems. In: Sergeyev YD, Kvasov DE, Astorino A, editors. Numerical Computations: Theory and Algorithms; Cham. Springer Nature Switzerland; 2025. p. 47–62

work page 2025
[6]

A quasi-steady-state minimum lap time simulation of race motor- cycles using experimental data

Biniewicz J, Pyrz M. A quasi-steady-state minimum lap time simulation of race motor- cycles using experimental data. Vehicle System Dynamics. 2024 Feb;62(2):372–394

work page 2024
[7]

Schwarz decomposition for parallel minimum lap-time problems: Evaluating against ADMM

Bartali L, Gabiccini M, Wright SJ. Schwarz decomposition for parallel minimum lap-time problems: Evaluating against ADMM. Vehicle System Dynamics. 2024 Sep;:1–26

work page 2024
[8]

A consensus-based alternating direction method of multipliers approach to parallelize large-scale minimum-lap-time problems

Bartali L, Grabovic E, Gabiccini M. A consensus-based alternating direction method of multipliers approach to parallelize large-scale minimum-lap-time problems. Multibody System Dynamics. 2024 Aug;61(4):481–507

work page 2024

Show all 28 references

[9]

How optimal is the minimum-time manoeuvre of an artificial race driver? Vehicle System Dynamics

Piccinini M, Taddei S, Pagot E, et al. How optimal is the minimum-time manoeuvre of an artificial race driver? Vehicle System Dynamics. 2024 Sep;0(0):1–28

2024
[10]

Statistical orbit determination

Tapley BD, Schutz BE, Born GH. Statistical orbit determination. Amsterdam: Elsevier Academic Press; 2004

2004
[11]

Chance-Constrained Optimal Path Planning With Obstacles

Blackmore L, Ono M, Williams BC. Chance-Constrained Optimal Path Planning With Obstacles. IEEE Transactions on Robotics. 2011 Dec;27(6):1080–1094

2011
[12]

Collision-free Motion Planning for Mobile Robots by Zero-order Robust Optimization-based MPC

Gao Y, Messerer F, Frey J, et al. Collision-free Motion Planning for Mobile Robots by Zero-order Robust Optimization-based MPC. In: 2023 European Control Conference (ECC); Jun.; Bucharest, Romania. IEEE; 2023. p. 1–6

2023
[13]

Robustified Time-optimal Point-to-point Motion Planning and Con- trol under Uncertainty

Zhang S, Swevers J. Robustified Time-optimal Point-to-point Motion Planning and Con- trol under Uncertainty. arXiv. 2025 Jan;arXiv:2501.14526

2025 arXiv
[14]

Robustified Time-optimal Collision-free Motion Plan- ning for Autonomous Mobile Robots under Disturbance Conditions

Zhang S, Bos M, Vandewal B, et al. Robustified Time-optimal Collision-free Motion Plan- ning for Autonomous Mobile Robots under Disturbance Conditions. In: 2024 IEEE In- 23 ternational Conference on Robotics and Automation (ICRA); May; Yokohama, Japan. IEEE; 2024. p. 14258–14264

2024
[15]

A simple and fast robust nonlinear model predictive control heuristic using n-steps-ahead uncertainty predictions for back-off calculations

Krog H, J¨ aschke J. A simple and fast robust nonlinear model predictive control heuristic using n-steps-ahead uncertainty predictions for back-off calculations. Journal of Process Control. 2024 Sep;141:103270

2024
[16]

Robust Trajectory Planning with Parametric Un- certainties

Brault P, Delamare Q, Giordano PR. Robust Trajectory Planning with Parametric Un- certainties. In: 2021 IEEE International Conference on Robotics and Automation (ICRA); May; Xi’an, China. IEEE; 2021. p. 11095–11101

2021
[17]

Trajectory Generation for Minimum Closed-Loop State Sensitivity

Giordano PR, Delamare Q, Franchi A. Trajectory Generation for Minimum Closed-Loop State Sensitivity. In: 2018 IEEE International Conference on Robotics and Automation (ICRA); May; Brisbane, QLD. IEEE; 2018. p. 286–293

2018
[18]

COP: Control & Observability-aware Planning ; 2022

B¨ ohm C, Brault P, Delamare Q, et al. COP: Control & Observability-aware Planning ; 2022

2022
[19]

Analysis for global characteristics of Lyapunov exponents in vehicle plane motion system

Meng F, Shi S, Zhang B, et al. Analysis for global characteristics of Lyapunov exponents in vehicle plane motion system. Scientific Reports. 2022 Jun;12(1):9300

2022
[20]

Use of Lyapunov Exponents to Predict Chaotic Vessel Motions

McCue LS, Troesch A W. Use of Lyapunov Exponents to Predict Chaotic Vessel Motions. In: Almeida Santos Neves M, Belenky VL, De Kat JO, et al., editors. Contemporary Ideas on Ship Stability and Capsizing in Waves. Vol. 97. Dordrecht: Springer Netherlands; 2011. p. 415–432

2011
[21]

Stability of Nonlinear, Time-Dependent Rotorcraft Systems Using Lyapunov Characteristic Exponents

Tamer A, Masarati P. Stability of Nonlinear, Time-Dependent Rotorcraft Systems Using Lyapunov Characteristic Exponents. Journal of the American Helicopter Society. 2016 Apr;61(2):1–12

2016
[22]

Rotorcraft stability analysis using Lyapunov char- acteristic exponents estimated from multibody dynamics

Cassoni G, Cocco A, Tamer A, et al. Rotorcraft stability analysis using Lyapunov char- acteristic exponents estimated from multibody dynamics. CEAS Aeronautical Journal. 2024 Jul;15(3):703–719

2024
[23]

Stability analysis of a nonlinear vehicle model in plane motion using the concept of Lyapunov exponents

Sadri S, Wu C. Stability analysis of a nonlinear vehicle model in plane motion using the concept of Lyapunov exponents. Vehicle System Dynamics. 2013 Jun;51(6):906–924

2013
[24]

Lyapunov matrix equation in system stability and control

Gaji´ c Z, Qureshi MTJ, editors. Lyapunov matrix equation in system stability and control. San Diego: Academic Press; 2010. (Mathematics in Science and Engineering; v. 195)

2010
[25]

Dynamical Systems and Geometric Mechanics: An Introduction

Maruskin JM. Dynamical Systems and Geometric Mechanics: An Introduction. Berlin/- Boston: De Gruyter, Inc; 2018. (De Gruyter Studies in Mathematical Physics Ser; v.48)

2018
[26]

Practical methods for approximate robust periodic optimal control of nonlinear mechanical systems [dissertation]

Gillis J. Practical methods for approximate robust periodic optimal control of nonlinear mechanical systems [dissertation]. Arenberg Doctoral School of Science, Engineering & Technology, Dept. of Electrical Engineering, KU Leuven; 2015

2015
[27]

CasADi: a software framework for nonlinear optimization and optimal control

Andersson JAE, Gillis J, Horn G, et al. CasADi: a software framework for nonlinear optimization and optimal control. Mathematical Programming Computation. 2019 Mar; 11(1):1–36

2019
[28]

On the implementation of an interior-point filter line-search al- gorithm for large-scale nonlinear programming

W¨ achter A, Biegler LT. On the implementation of an interior-point filter line-search al- gorithm for large-scale nonlinear programming. Mathematical Programming. 2006 Mar; 106(1):25–57. 24

2006

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A free-trajectory quasi-steady-state optimal-control method for minimum lap-time of race vehicles

Veneri M, Massaro M. A free-trajectory quasi-steady-state optimal-control method for minimum lap-time of race vehicles. Vehicle System Dynamics. 2020 Jun;58(6):933–954

work page 2020

[2] [2]

Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles

Lovato S, Massaro M. Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles. Vehicle System Dynamics. 2022 Nov;60(11):3650–3667

work page 2022

[3] [3]

A three-dimensional free-trajectory quasi-steady-state optimal- control method for minimum-lap-time of race vehicles

Lovato S, Massaro M. A three-dimensional free-trajectory quasi-steady-state optimal- control method for minimum-lap-time of race vehicles. Vehicle System Dynamics. 2022 May;60(5):1512–1530

work page 2022

[4] [4]

Comparison of direct and indirect methods for minimum lap time optimal control problems

Dal Bianco N, Bertolazzi E, Biral F, et al. Comparison of direct and indirect methods for minimum lap time optimal control problems. Vehicle System Dynamics. 2019 May; 57(5):665–696

work page 2019

[5] [5]

A Direct/Indirect Approach to Optimal Control Problems

Bertolazzi E, Biral F. A Direct/Indirect Approach to Optimal Control Problems. In: Sergeyev YD, Kvasov DE, Astorino A, editors. Numerical Computations: Theory and Algorithms; Cham. Springer Nature Switzerland; 2025. p. 47–62

work page 2025

[6] [6]

A quasi-steady-state minimum lap time simulation of race motor- cycles using experimental data

Biniewicz J, Pyrz M. A quasi-steady-state minimum lap time simulation of race motor- cycles using experimental data. Vehicle System Dynamics. 2024 Feb;62(2):372–394

work page 2024

[7] [7]

Schwarz decomposition for parallel minimum lap-time problems: Evaluating against ADMM

Bartali L, Gabiccini M, Wright SJ. Schwarz decomposition for parallel minimum lap-time problems: Evaluating against ADMM. Vehicle System Dynamics. 2024 Sep;:1–26

work page 2024

[8] [8]

A consensus-based alternating direction method of multipliers approach to parallelize large-scale minimum-lap-time problems

Bartali L, Grabovic E, Gabiccini M. A consensus-based alternating direction method of multipliers approach to parallelize large-scale minimum-lap-time problems. Multibody System Dynamics. 2024 Aug;61(4):481–507

work page 2024