An Explicit Discrete-Time Dynamic Vehicle Model with Assured Numerical Stability

Bin Zhao; Guojian Zhan; Haoyu Gao; Qiang Ge; Shengbo Eben Li; Yuming Yin

REVIEW 2 major objections 5 minor 17 references

An Explicit Discrete-Time Dynamic Vehicle Model with Assured Numerical Stability

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An explicit discrete-time dynamic vehicle model stays numerically stable at zero speed.

desk verdict A useful semi-implicit discretization of the single-track model that stays finite at zero speed, with a stability theorem that needs a strict-inequality fix and a Condition 1 that is only empirically checked. read the letter →

arxiv 2411.17334 v1 pith:TTEEBYU2 submitted 2024-11-26 eess.SY cs.SY

Guojian Zhan , Qiang Ge , Haoyu Gao , Yuming Yin , Bin Zhao , Shengbo Eben Li This is my paper · ORCID

classification eess.SYcs.SY

keywords dynamicvehiclemodelnumericalstabilitylow-speedsingularityexplicitdiscretizationbackwardEulerlineartirepredictivecontrolstop-startdriving

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 aims to establish that a six-state single-track dynamic vehicle model can be written in an explicit discrete-time form—the next state is a direct function of the current state and control input—without experiencing the low-speed singularity that makes conventional dynamic models explode near standstill. The singularity comes from tire slip-angle formulas whose denominator is the longitudinal speed $U$; the proposed scheme removes that source of blow-up while retaining the explicit form that model-based controllers prefer. The central guarantee is that the update for lateral velocity $V$ and yaw rate $\omega$ is obtained through a backward-Euler-style linear solve that becomes algebraic under the linear tire model, and the paper proves a sufficient condition for numerical stability. Simulation comparisons with a commercial full-vehicle simulator and real-vehicle stop-start experiments are presented as evidence that the model is accurate and stable in exactly the low-speed regime where earlier explicit models fail.

What carries the argument

The load-bearing mechanism is a mixed discretization: $X$, $Y$, $\varphi$, and $U$ are advanced by forward Euler, while $V_{k+1}$ and $\omega_{k+1}$ are written as backward-Euler-style updates in which the unsolved variable is the one being updated. With the linear tire model, this pair of equations collapses to a two-by-two linear solve, so the model stays explicit. That solving step moves the cornering stiffnesses into the denominators of the lateral-velocity and yaw-rate updates, which is what removes the pure speed dependence and kills the singularity. The error-propagation analysis then factors each step into a block matrix whose two-by-two submatrix is $\hat{A}_k$, and Condition 1 demands $\|\hat{A}_k\| \le 1$, a sufficient bound for the product of these matrices to remain bounded.

What would settle it

Run two trajectories from nearby initial states under identical control for a vehicle parameter set or a step size where the norm of the two-by-two error-amplification submatrix exceeds 1, and observe the state difference growing without bound; alternatively, construct a trajectory where the norm equals 1 at every step and show the accumulated error grows linearly, which would violate the boundedness that Proposition 5.3 claims.

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Extended reading notes

Core claim

The paper's central claim is that its update rule is an explicit discrete-time dynamic vehicle model whose lateral-velocity and yaw-rate updates have denominators $mU_k - T_s(k_f + k_r)$ and $I_z U_k - T_s(l_f^2 k_f + l_r^2 k_r)$, rather than just $U_k$. Because these denominators do not vanish at $U=0$, the slip-angle blow-up that characterizes the low-speed singularity is avoided, and the model remains explicit, so no fixed-point iteration is needed at a step. The paper states that if Condition 1 holds—the two-by-two error-amplification submatrix of the Jacobian has norm at most 1 at every intermediate point—then Proposition 5.3 guarantees numerical stability in the sense that two trajectories driven by the same control sequence stay within a bounded distance forever. The supporting evidence is numerical verification of the condition for speeds from 0 to 25 m/s at step sizes up to 0.1 s, together with accuracy comparisons and real-vehicle trajectories.

Load-bearing premise

The stability theorem assumes Condition 1, that the two-by-two error-amplification submatrix has norm at most 1 over the whole operating domain; the paper only checks this numerically for one vehicle prototype, three step sizes, and speeds up to 25 m/s, and the proof's bounding step leaves the exactly-equal-to-1 case unresolved.

Editorial extensions

If this is right

The model works where earlier explicit dynamic models break down: urban stop-start maneuvers with $U$ crossing zero no longer produce exploding lateral states.
Because the update is explicit and algebraic, predictive controllers and model-based learning rollouts can call it without iterative solves at each step.
If Condition 1 holds, initial-state errors stay bounded over an arbitrarily long horizon, so small sensing or localization errors do not compound without limit.
The full-vehicle-simulator comparisons show trajectory errors typically an order of magnitude smaller than the kinematic model's, so the added dynamic fidelity comes without a stability penalty.
Stability at step sizes up to $0.1$ s means longer prediction horizons are available under fixed computational budgets.

Reading between the lines

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

Beyond the paper: the stability condition is verified numerically, not derived; a useful extension would be an explicit inequality relating $T_s$, mass, inertia, and cornering stiffness that guarantees the norm bound without simulation.
Beyond the paper: since the discrete map is explicit and smooth away from the singular limit, differentiating through it for gradient-based control or learned dynamics is straightforward, but whether the stability margin survives learned parameter changes is untested.
Beyond the paper: at $U=0$ the update for $V$ and $\omega$ becomes a fixed linear map independent of the control; analyzing that map's eigenvalues would reveal whether a fully stopped vehicle can accumulate spurious yaw rate over many idle steps.

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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

Here is the short version: this is a workmanlike engineering paper with a genuinely convenient result — an explicit discrete-time single-track model with linear tires whose V and omega updates have denominators mU - Ts(kf+kr) and IzU - Ts(lf^2 kf + lr^2 kr), so they are finite at U=0. The trick is to apply backward-Euler-like updates to V and omega, solve them in closed form, and use forward Euler for X, Y, phi, U. The algebra checks out, and the simulations against CarSim show much better accuracy than the kinematic model and better stability than forward Euler at larger step sizes. That part is solid and useful for MPC and learning-based control in stop-start scenarios.

The soft spots are in the stability guarantee. Proposition 5.3's proof has a genuine gap at the boundary ||A*|| = 1. Equation (24)'s geometric series only works for ||A*|| < 1; the paper's handling of the equality case in (26) simply asserts ||b_{k-1..0}|| <= ||b*||, but the correct bound is k||b*||, which grows without bound. So the theorem, as proved, establishes bounded disturbance amplification only under the strict inequality. That's probably the operative case in practice — their Figure 3 shows the norm well below 1 for the tested range — but the stated Condition 1 says <= 1. The fix is easy: state Condition 1 with '< 1', or give a separate argument for the boundary. This matters because 'assured numerical stability' is the paper's headline claim.

Condition 1 itself is not derived from the parameters; it's verified numerically for one CarSim prototype, three step sizes, and U in 0-25 m/s. That's honest and the empirical coverage is reasonable, but it falls short of 'assured' for a general vehicle. One stress-test note claims the denominators also vanish at positive speeds; that's wrong — with negative cornering stiffnesses the zero-crossing speeds are negative, so they're outside the stated operating domain. No point holding that against the paper. The other stress-test points land.

The real-vehicle experiments are open-loop with RMS errors in the 0.3-2.6 m range. They show the model tracks trajectories, but there's no baseline comparison, so they don't demonstrate superiority — they demonstrate adequacy. Fine for a practical note, weaker than the simulation claims.

Overall: this deserves a serious referee and likely acceptance after a moderate revision. The core model is useful and correct; the proof needs the inequality fix and the operating domain needs to be stated explicitly in Condition 1. I'd take it to reading group and would cite it if I were working on predictive control for urban driving.

Referee Report

2 major / 5 minor

Summary. This paper proposes an explicit discrete-time dynamic single-track vehicle model. The discretization uses forward Euler for position, heading, and longitudinal speed, and a backward-Euler-inspired but algebraically solved update for lateral speed and yaw rate, yielding the closed-form map (12). The main claims are (i) the model avoids the low-speed singularity that afflicts standard dynamic models because the denominators of the lateral updates contain cornering-stiffness terms in addition to U, and (ii) the model is numerically stable in the sense of bounded error propagation, under a sufficient condition (Condition 1) on the norm of a Jacobian submatrix. The paper proves that condition is sufficient, verifies it numerically for one CarSim prototype over 0–25 m/s and three step sizes, compares accuracy against a kinematic model and stability against forward-Euler discretization, and reports open-loop real-vehicle experiments.

Significance. If the stability guarantee were fully established, this would be a practically valuable contribution to model-based control at low speed and in stop-start scenarios. The paper has clear strengths: the explicit form is genuinely useful for MPC and rollout, the avoidance of a U-only denominator is a concrete design idea, the algebraic derivation of (12) is transparent, and the empirical comparison and real-vehicle tests are extensive. However, the central 'assured numerical stability' claim is currently only a conditional theorem, and the proof has a gap at the boundary of Condition 1. The numerical validation covers a single parameter set and only verifies the non-strict inequality, so the theoretical guarantee is not as strong as advertised.

major comments (2)

[§5.3, Eq. (26)] In the proof of Proposition 5.3, the bound in (24) is obtained by summing a geometric series and is valid only when ||Â*|| < 1. The text then handles the case ||Â*|| = 1 via (26), asserting ||b_{k−1..0}|| ≤ ||b*||, but this does not follow from (23). The correct bound is ||b_{k−1..0}|| ≤ k||b*||, which is unbounded as k→∞. Consequently, Proposition 5.3 is not proved under Condition 1 as stated (≤ 1); it is proved only under the strict inequality ||Â_k(τ_k)|| < 1. Since the paper's title and abstract claim 'assured numerical stability', this gap is load-bearing and must be repaired, for example by strengthening Condition 1 to strict contraction with a margin and verifying that margin, or by supplying a refined argument for the equality case that rules out linear growth of the accumulated b-term.
[§6.2, Figure 3] The empirical verification in Section 6.2 checks ||Â_k(τ_k)|| ≤ 1 for a single CarSim parameter set (Table 1), step sizes 0.001, 0.01, and 0.1 s, and longitudinal speeds in [0, 25] m/s. This does not close the gap identified in Proposition 5.3, because the proof requires the strict inequality < 1, while the verification reports only non-strict ≤ 1. Moreover, because only one vehicle/parameter set is tested, the general claim of 'assured numerical stability' for the proposed model is not established across parameter variations. The authors should either prove an analytic condition ensuring ||Â|| < 1 for a parameterized family, or explicitly restate the contribution as a model whose stability is verified conditionally/numerically, with the strict-inequality issue resolved.

minor comments (5)

[§2.3 / Lemma 5.2] In Lemma 5.2, the statement writes g'(m) = (f(d)−f(c))/(d−c); the right-hand side should use g, not f.
[§1, Contributions (4)] The phrase 'strongly evidents' should read 'strongly evidences'.
[§4.1] The description 'substituting local (vehicle-attach) coordinate for ground coordinate system' is unclear; please spell out why setting a→U̇ is justified.
[§5.3] The sentence 'using the nature of the inverse proposition' is vague; the argument that bounded norms imply no infinite elements should be stated formally.
[§5.1] Definition 5.1 uses lim_{k→∞} ||x_k−y_k|| ≤ C; if the sequence does not converge, the limit may not exist. Consider replacing the limit with limsup or explicitly require convergence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the model derivation and the conditional stability theorem are self-contained, and the flagged proof gap at ||A*||=1 is a rigor issue, not a circular reduction.

full rationale

The paper derives its explicit discrete-time model (12) by mixing forward Euler updates for X, Y, phi, U with backward-Euler-style updates for V and omega. The claimed avoidance of low-speed singularity follows directly from the denominators mU_k - T_s(k_f+k_r) and I_z U_k - T_s(l_f^2 k_f + l_r^2 k_r), which remain nonzero at U=0 for negative cornering stiffnesses; this is a construction property, not a conclusion imported from the stability theorem. The stability analysis is explicitly conditional: Condition 1 states ||A_hat_k(tau_k)|| <= 1, and Proposition 5.3 proves bounded error propagation assuming that condition. Verifying Condition 1 numerically in Section 6.2 for one CarSim prototype, three step sizes, and speeds 0-25 m/s is an empirical check of a stated sufficiency condition, not a fitted parameter being relabeled as a prediction. The paper itself flags the case ||A*|| = 1 in the proof of Proposition 5.3 and asserts a bound (26) that is not justified by the preceding inequality; this is an admitted mathematical gap and a correctness risk, but it is not a circular step because the theorem's conclusion is not baked into its assumptions. Self-citations in the reference list (e.g., [2], [13]) are background and are not load-bearing for the model derivation or the stability claim. Real-vehicle experiments use parameters identified in preliminary trials and compare open-loop simulations against measured trajectories; this is validation rather than circular prediction. No step in the derivation chain reduces to its own inputs, so the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a linear tire model, a modified longitudinal update, and a numerically verified matrix-norm condition. The tire stiffnesses are fitted to a CarSim prototype or real-vehicle trials. No new physical entities are introduced.

free parameters (2)

Front axle cornering stiffness k_f = -128916 N/rad (simulation), -186000 N/rad (real vehicle)
Identified from CarSim prototype or from preliminary real-vehicle trials (Tables 1 and 3). The stability condition and the model outputs depend on these values.
Rear axle cornering stiffness k_r = -85944 N/rad (simulation), -183000 N/rad (real vehicle)
Identified from CarSim prototype or from preliminary real-vehicle trials (Tables 1 and 3). The stability condition and the model outputs depend on these values.

assumptions (4)

domain assumption Linear tire force model with small-angle approximation: F_f = k_f((V + l_f*omega)/U - delta), F_r = k_r((V - l_r*omega)/U).
Invoked in equation (2) and is essential for the closed-form updates in (12). Valid only for small slip angles and moderate lateral acceleration.
ad hoc to paper Longitudinal dynamics simplification: drop V*omega and (1/m)F_f*sin(delta) from the U update, so U_{k+1} = U_k + T_s*a_k.
Stated in Section 4.1 as neglecting longitudinal resistance and substituting local coordinates. This changes the model from the original continuous dynamics and is not validated against the original ODE.
ad hoc to paper Condition 1: ||A_hat_k(tau_k)|| <= 1 for all k and all intermediate interpolation points.
Proposition 5.3 shows stability if this matrix-norm condition holds uniformly, but the condition is never proven from the model; it is only verified numerically in Section 6.2 for one vehicle, one speed range, and three step sizes.
domain assumption State trajectories remain within a closed bounded domain [x_min, x_max] for all time.
The stability proof defines maxima of b and A_hat over this closed domain in (25); the proof does not establish that the model keeps states inside the domain.

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Cite this review

Pith. "Pith review of An Explicit Discrete-Time Dynamic Vehicle Model with Assured Numerical Stability." pith.science (2026). https://pith.science/paper/TTEEBYU2

@misc{pith2026241117334,
  author       = {Pith},
  title        = {Pith review of: An Explicit Discrete-Time Dynamic Vehicle Model with Assured Numerical Stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTEEBYU2}},
  note         = {Machine review of arXiv:2411.17334}
}

read the original abstract

Numerical stability is of great significance for discrete-time dynamic vehicle model. Among the unstable factors, low-speed singularity stands out as one of the most challenging issues, which arises from that the denominator of tire side angle term only contains the vehicle longitudinal speed. Consequently, for the common low-speed and stop-start driving scenarios, the calculated tire slip angle will approach infinity, which will further lead to the numerical explosion of other vehicle states. In response to this critical challenge, we propose a discrete-time dynamic vehicle model that effectively mitigates the low-speed singularity issue, ensuring numerical stability and maintaining the explicit form-highly favored by model-based control algorithms. To validate the numerical stability of our model, we conduct a rigorous theoretical analysis, establishing sufficient conditions for stability, and conduct extensive empirical validation tests across a wide spectrum of speeds. Subsequent to the validation process, we conduct comprehensive simulations comparing our proposed model with both kinematic models and existing dynamic models discretized through the forward Euler method. The results demonstrate that our proposed model shows better comprehensive performance in terms of both the accuracy and numerical stability. Finally, the real vehicle experiments are carried out to support that our proposed model can closely aligns to the real vehicle trajectories showcasing its practicality and ease of use. Notably, our work stands as the pioneering endeavor in introducing an explicit discrete-time dynamic vehicle model suitable for common urban driving scenarios including low-speed and stop-start.

Figures

Figures reproduced from arXiv: 2411.17334 by the authors.

**Figure 1.** Vehicle model 2.3. Classical discretization methods In order to apply the ordinary differential equation model to design controller at digital computers, it is necessary to implement numerical discretization. As typical representatives of the explicit method and the implicit method, forward Euler method and backward Euler method are presented as shown in (4a) and (4b), respectively. Where footnote 𝑘 is the time step… view at source ↗

**Figure 2.** The physical meaning of numerical stability definition. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** The second norm of the sub-coefficient matrix when the longitudinal vehicle speed ranges from 0 to 25 m/s. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Time efficiency benchmark Secondly, robustness against disturbances or variations is also important for a realiable vehicle model. Here we conduct a robustness experiment to test whether our model can simulate realistic trajectory with the noisy input common seen in re…

**Figure 5.** Figure 5: Robustness test on resisting noise. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: Lateral states with varying discretization steps under two step steering input. 𝑈0 = 8 m/s, the first step steering input 𝛿1 = 0.1347 rad at 0.0 s, and the second steering input 𝛿2 = 0.2674 rad at 1.0 s. It can be seen from [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Trajectory of the close-loop control process of a stop-start driving scenario. The obstacle changes position from (15, 15) to (18, 12) when the time is 3.1 s. 0 1 2 3 4 5 6 7 8 Time (s) 0 2 4 6 8 U ( m / s ) (a) Longitudinal velocity 0 1 2 3 4 5 6 7 8 Time (s) -4 -2 0 …

**Figure 8.** Figure 8: Control input of the closed-loop control experiment. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Hardwares in real vehicle experiments [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: Longitudinal real vehicle experiments 0.0 2.5 5.0 7.5 10.0 12.5 Time [s] 0 2 4 Velocity [m/s] Real Sim (a) Lat-1 longitudinal velocity 0 10 20 30 40 X location [m] 15 10 5 0 Y location [m] (b) Lat-1 trajectory 0.0 2.5 5.0 7.5 10.0 12.5 Time [s] 0.0 2.5 5.0 7.5 Velocit…

**Figure 11.** Figure 11: Lateral real vehicle experiments 8. Conclusion In this paper, we propose an explicit discretized dynamic vehicle model and provide a sufficient condition that guarantees its numerical stability. Our proposed model outperforms forward21 [PITH_FULL_IMAGE:figures/full_…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Vehicle dynamics applications of optimal control theory.Vehicle System Dynamics, 49(7):1073–1111, 2011

RS Sharp and Huei Peng. Vehicle dynamics applications of optimal control theory.Vehicle System Dynamics, 49(7):1073–1111, 2011

work page 2011

[2] [2]

Continuous-time policy optimization

Guojian Zhan, Yuxuan Jiang, Jingliang Duan, Shengbo Eben Li, Bo Cheng, and Keqiang Li. Continuous-time policy optimization. In 2023 American Control Conference (ACC), pages 3382–

work page 2023

[3] [3]

An overview on vehicle dynamics

Shaopu Yang, Yongjie Lu, and Shaohua Li. An overview on vehicle dynamics. International Journal of Dynamics and Control, 1:385–395, 2013

work page 2013

[4] [4]

Validation of vehicle dynamics simulation models–a review

Emir Kutluay and Hermann Winner. Validation of vehicle dynamics simulation models–a review. Vehicle System Dynamics, 52(2):186–200, 2014

work page 2014

[5] [5]

Camilo Andr ´es Manrique-Escobar, Carmine Maria Pappalardo, and Domenico Guida. On the analytical and computational methodologies for modelling two-wheeled vehicles within the multi- body dynamics framework: a systematic literature review.Journal of Applied and Computational Mechanics, 8(1):153–181, 2022

work page 2022

[6] [6]

Kinematic and dynamic ve- hicle models for autonomous driving control design

Jason Kong, Mark Pfeiffer, Georg Schildbach, and Francesco Borrelli. Kinematic and dynamic ve- hicle models for autonomous driving control design. In2015 IEEE Intelligent Vehicles Symposium (IV), pages 1094–1099, Seoul, Korea, 2015. IEEE

work page 2015

[7] [7]

Nu- merical methods in vehicle system dynamics: state of the art and current developments

Martin Arnold, Bernhard Burgermeister, Claus F¨ uhrer, Gerhard Hippmann, and Georg Rill. Nu- merical methods in vehicle system dynamics: state of the art and current developments. Vehicle System Dynamics, 49(7):1159–1207, 2011

work page 2011

[8] [8]

A discussion on euler method: A review

BN Biswas, Somnath Chatterjee, SP Mukherjee, and Subhradeep Pal. A discussion on euler method: A review. Electronic Journal of Mathematical Analysis and Applications , 1(2):2090– 2792, 2013

work page 2013