Real-Time Mass and Axle Load Estimation in Multi-Axle Trucks Through Fusion of TPMS Pressure and Vision-Derived Tire Deformation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper presents a real-time mass and axle-load estimation method for multi-axle trucks by fusing TPMS pressure and vision-derived tire deformation data. The adaptive Kalman filter fusion approach is innovative, achieving a relative mass estimation error below 5%. However, the study has several limitations, including  a single tire test (315/80 R22.5), and missing real-world deployment images. These issues may affect the method's generalizability and practical applicability.

1. Missing Experimental Setup Images: No figures depict the vision system, TPMS integration, or tire deformation measurement. Visual documentation is essential for reproducibility.
2. Only 315/80 R22.5 tires are tested. Does the method generalize to other sizes (e.g., 295/75 R22.5, 385/55 R22.5)? A discussion or additional tests are needed.
3. The claimed runtime (<33 ms) lacks hardware specifications (CPU, camera fps). Specify the hardware used and discuss scalability to low-cost embedded systems.
4. Is the 30 Hz update rate sufficient for dynamic load shifts (e.g., braking, uneven terrain)? A sensitivity analysis on motion dynamics would strengthen the results.
5. The single-degree-of-freedom oscillator model may oversimplify tire-road interactions. The relevant content requires further explanation.

Author Response

Comment: Missing Experimental Setup Images.
We acknowledge that the manuscript currently lacks figures of the vision system, TPMS integration, and tire deformation measurement. While this study is theoretical and simulation-based, we agree that visual documentation enhances reproducibility. In a revised version, we will include schematics of the camera positioning, TPMS sensor configuration, and the deformation measurement workflow. Additionally, we plan to add conceptual diagrams that illustrate the fusion pipeline from inputs to outputs.

Comment: Tire Size Generalization.
The initial validation was performed with 315/80 R22.5 tires as a baseline, as these are widely used in European long-haul trucks. However, the model formulation explicitly parameterizes tire stiffness, damping, and contact patch geometry, which allows extension to other sizes (e.g., 295/75 R22.5, 385/55 R22.5). We will expand the discussion to highlight this scalability and provide interpolated parameter ranges from published tire datasets. Future work will include numerical validation across multiple tire geometries.

Comment: Runtime and Hardware Specifications.
The reported runtime (<33 ms per update) was obtained on a standard Intel i7-12700 CPU with 16 GB RAM, and vision preprocessing simulated at 60 fps. We will explicitly specify the hardware configuration and discuss scaling strategies for low-cost embedded systems. Preliminary profiling suggests that with optimized C++ implementations and GPU acceleration, the algorithm can be deployed on automotive-grade processors such as NVIDIA Jetson or ARM-based SoCs.

Comment: Update Rate and Dynamic Loads.
The current 30 Hz update rate was chosen to balance estimation stability and computational feasibility. We agree that this assumption is valid primarily for quasi-static and moderately dynamic scenarios (road grade <1%, vertical acceleration $|\ddot{z}|<0.2 g$). For highly dynamic cases (braking, uneven terrain), higher update rates or adaptive filtering are required. We will include a sensitivity analysis that shows performance degradation with dynamic load shifts and explain how the adaptive Kalman filter compensates under these conditions. Nevertheless, the objective of this invention is to estimate the weight in static way with error lower than 5% to comply with further regulation.

 

Comment: Tire Model Complexity.
The single-degree-of-freedom oscillator representation was selected for tractability and its compatibility with TPMS + deformation inputs. We agree that this abstraction may omit higher-order tire–road effects (e.g., lateral stiffness, nonlinear damping). We will clarify this limitation and position the model as a first-order approximation suitable for mass estimation, while noting that extensions to higher-fidelity models (finite element or nonlinear lumped-parameter) are part of future research.

 

Reviewer 2 Report

Comments and Suggestions for Authors

The study is based on the author's patent: "Gallego, J. WEIGHTING SYSTEM AND WEIGHTING METHOD FOR A VEHICLE. EP 4071449231 A1, 10 2022. European Patent application", Patent EP4071449A1, which fuses TPMS pressure and radial deformation from a chassis camera.

As the author puts it, "The present study extends that idea to rigid and articulated trucks with up to four axles." The main contribution of the work is apparently in the presented graphs, which compare the dependence of radial deformation and footprint area on the estimated mass for a different number of vehicle axles.

Further, the author writes that the work publishes a mathematical model from his patent:
"This paper derives the mathematical model from the original patent and describes main parameter identification."

This is a good idea for a scientific article, but at the same time, the article as it is, is still written very superficially.

The author promises something interesting: "it outlines integration with intelligent transport systems", however, nothing of the sort is actually described in the article. Where are the architectural diagrams with the sequence of actions for integrating his solution into industry (for example, at mass measurement posts) or explicit algorithms for how to use all of this? What equipment exactly should be used?

The author integrates differential equation approximation and Kalman-type filters into the model. What needs to be shown here so that it is understandable to engineers is a Simulink / XCos-type diagram and then tell how the author moved from it to his Python code.

Where is the description of the parameters of the test trucks and the dataset used? Usually this is described in the "Data showcase" section, where the data types, their examples are described and a link to the dataset is given.

The fixed-step RK4 algorithm (line 132) is in any textbook. At the same time, there are several approximation algorithms in the same Simulink / XCos and one also needs to be able to choose and prove why this particular one was used.

The text has not been checked. Examples of obvious errors:
cammera (Fig. 1)
::contentReference[oaicite:0]index= (line 161)

Another very important point. The author claims a patent, but no similar (competing) developments are cited in the article. It is strange, such an analysis was needed for the patent and it should have been cited (with an emphasis on the academy) in the article in the "Related work" section.

In general, the article is still too raw for publication.

Author Response

Dear Reviewer,

Thank you for your careful and detailed review of my manuscript. I appreciate the constructive feedback, which has allowed me to significantly improve the clarity and completeness of the paper. Below I summarize how each of your concerns has been addressed in the revised version:

1. Connection to the patent and novelty: The manuscript now explicitly explains in the Introduction that the study builds upon Patent EP4071449A1 and extends the concept to rigid and articulated trucks with up to four axles. A dedicated paragraph also highlights how the present work goes beyond TPMS-only, vision-only, and indirect mass-estimation approaches.

2. Integration with intelligent transport systems: The Discussion and Conclusions now include a clear deployment scenario. The estimator is described as running on a gateway ECU, fusing TPMS and camera inputs at 30–60 Hz, with outputs directed to overload alerts, telematics platforms, V2X communication for road-user charging and WIM pre-screening, and adaptive driver-assistance systems. Message cadence, metadata, and privacy considerations are also specified.

3. Mathematical model and implementation: The Materials and Methods section provides the full derivation of the mathematical model, including the state-space formulation, equations of motion, and the adaptive Kalman filter scheme. The transition to numerical implementation is described step by step, and the Runge–Kutta integration scheme is written explicitly in LaTeX and programmed in python.

4. Data and test parameters: A Data Showcase section has been added in Materials and Methods, including tables of tire stiffness, damping, and contact-pressure parameters at different inflation levels, as well as uncertainty ranges used in the Latin Hypercube sampling. This provides a clear dataset description for reproducibility.

5. Justification of the RK4 method: The Numerical Implementation section now explicitly justifies the use of fixed-step RK4. It explains that RK4 provides a balance between stability and computational efficiency under real-time constraints, and that RK45 was also tested with negligible accuracy gains but higher runtime variability.

6. Language and corrections: Typographical errors such as “cammera” have been corrected, and formatting issues such as spurious references (::contentReference) have been removed. The text has undergone a full language revision for correctness and clarity.

7. Related work and competing developments: The Introduction and reference list now include citations to similar and competing approaches in the academic literature, including recent studies on tire deformation analysis, pressure-dependent stiffness modeling, and Bayesian fusion for vehicle mass estimation. This situates the novelty of the present work more clearly within the field.

Finally, I would like to note respectfully that the journal has published several articles in the same area that are considerably less detailed than the present submission, particularly in terms of mathematical derivations, uncertainty analysis, and numerical validation. The revised manuscript now includes a complete dynamical model, explicit differential equations, block-diagonal Kalman filter formulation, sensitivity analysis (PRCC and Sobol), and integration with ITS applications. In this sense, I believe the current version already exceeds the methodological depth of comparable articles published in Inventions.

I sincerely thank you again for your constructive comments. I hope you will find that the revised version addresses your concerns and demonstrates the significance and originality of this research.

Reviewer 3 Report

Comments and Suggestions for Authors

   The manuscript proposes a theoretical and numerical framework for estimating the gross mass and axle loads of multi-axle trucks by fusing tyre pressure monitoring system (TPMS) data with vision-based tyre deformation measurements. The study develops a pressure-dependent dynamic tyre model, introduces two independent force surrogates, and integrates them via an adaptive Kalman filter. Numerical validation suggests that the method can achieve sub-5% mass estimation error using only standard on-board sensors, with robustness confirmed through uncertainty analysis. The paper addresses a relevant regulatory and engineering problem and demonstrates clear potential for practical application. However, the work is purely theoretical and requires substantial revisions to strengthen its contribution.

1. The framework is fully theoretical and relies only on literature-derived parameters. While the simulation results are encouraging, experimental validation is essential to confirm feasibility in real-world conditions. At minimum, a discussion on how the model would be tested on actual vehicles, including possible sources of error such as tyre wear, road irregularities, or environmental conditions, should be expanded.
2. The paper claims several contributions (pressure-dependent compliance model, contact patch mapping, fusion scheme). However, these are incremental extensions of prior work (e.g., the cited patent). The novelty needs to be highlighted more explicitly, particularly in contrast to existing TPMS–vision fusion methods and other indirect mass estimation approaches.
3. The abstract and conclusion mention integration with intelligent transport systems, but no clear path or scenario is described. The authors should elaborate on how the proposed method would support applications such as load monitoring, traffic safety, or road-user charging in ITS environments.
4. The limitations are briefly acknowledged in the conclusion, but a deeper discussion is required. The authors should explicitly address model assumptions (e.g., quasi-static conditions, linear approximations) and their implications for dynamic or extreme driving scenarios. Future work should be outlined in greater detail, including experimental design, potential extension to lateral dynamics, or the use of advanced fusion techniques.
5. The background section should better highlight the broader significance of real-time mass and axle-load estimation. In particular, accurate load information is not only relevant for compliance and safety but also plays a crucial role in estimating the tire-road friction coefficient, which directly impacts vehicle control strategies. Expanding this discussion would help underline the importance of the present work for advanced control applications. Recent study “An Innovative Robust H∞ Adaptive Cruise Control Method with Input Constraint Considering Tire-road Friction Coefficient, 2023 IEEE 26th International Conference on Intelligent Transportation Systems (ITSC), Bilbao, Spain, 2023, pp. 3659-3666” can be cited to strengthen the motivation of the paper.

Author Response

Comment: Experimental validation and real-world error sources

We acknowledge that the present study is theoretical and parameterized from the literature. In the revised manuscript we will add comments for “Experimental Validation Plan,” indicating a theoretical protocol for on-vehicle testing: (i) instrumentation (production TPMS and a calibrated monocular camera), (ii) time-synchronization and extrinsic calibration, (iii) reference measurements via portable axle scales or an instrumented WIM segment, and (iv) a test matrix spanning load levels, inflation pressures, tyre temperature, asphalt types, and mild grades. We will also specify field error sources (tyre wear and temperature drift (affecting kr(P)k_r(P)kr​(P) and cr(P)c_r(P)cr​(P)), road roughness, TPMS bias/latency, and vision reconstruction error for δ\deltaδ) and we will state acceptance criteria a priori, e.g., mean absolute error <5% for gross mass and for each axle; 95th percentile <7%. 

Comment: Novelty relative to prior work (including the cited patent)

We will sharpen the contribution statement and add a comparison table clarifying how our approach differs from TPMS-only, vision-only, and other indirect mass-estimation methods, as well as from the patent cited in the paper. The revised text will highlight:
• The explicit fusion of two independent force surrogates (stiffness-based Fk=kr(P) δF_k=k_r(P)\,\deltaFk​=kr​(P)δ and pressure–area Fq=q(P) S(δ)F_q=q(P)\,S(\delta)Fq​=q(P)S(δ)) within an adaptive Kalman filter;
• covariance scheduling driven by the integrator’s local error estimate, linking numerical uncertainty (RK step error) to model confidence;
• piecewise-linear constitutive maps kr(P), cr(P), q(P)k_r(P),\,c_r(P),\,q(P)kr​(P),cr​(P),q(P) with confidence intervals and uncertainty propagation;
• an observability/identifiability analysis for the multi-axle architecture;
• linear-in-wheels computational complexity via block-diagonal per-wheel filtering;
• global sensitivity with PRCC plus variance-based Sobol indices and posterior intervals for MMM and Faxle​.

Comment: Path to Intelligent Transport Systems (ITS) integration

We will extend the abstract and conclusion with a concrete deployment scenario. The estimator runs at the edge (gateway ECU): inputs Pi(t)P_i(t)Pi​(t) from the TPMS and δi(t)\delta_i(t)δi​(t) from the vision module are fused at 30–60 Hz; outputs are wheel and axle loads and gross mass M(t). These outputs feed (i) on-board alerts for over-load and imbalance, (ii) telematics to back-office platforms for compliance and predictive maintenance, (iii) V2X messages to support road-user charging and weigh-in-motion pre-screening, and (iv) adaptive chassis/driver-assistance functions that benefit from up-to-date load estimates. We will describe the message cadence, minimal metadata, and privacy considerations.

 

Comment: Limitations, assumptions, and future work

We will add “Assumptions and Validity Domain” to state explicitly: level-road and quasi-static bounds (e.g., ∣z¨∣<0.2g|\ddot z|<0.2g∣z¨∣<0.2g, grade <1%), piecewise linearization of kr,cr,qk_r,c_r,qkr​,cr​,q in the 100–300 kPa range, SDOF radial model without lateral coupling, and constant unsprung masses. We will include a short sensitivity study where these assumptions are deliberately violated (braking pulses, moderate undulations) and report how performance degrades and when the full dynamic model or a higher update rate is required. The Future Work section will detail: (i) on-vehicle experiments and dataset release, (ii) extension to lateral dynamics and load transfer, (iii) richer constitutive laws outside the nominal pressure range, and (iv) Bayesian or EM-based adaptation for process/measurement noise.

 

Comment: Broader significance: mass, axle loads, and friction-aware control

We agree that real-time mass and axle loads matter beyond compliance and safety. They directly shape normal loads and thus the feasible friction ellipse, improving on-line estimation of tyre–road friction and enabling better parameterization of control strategies (e.g., ACC/MPC, ESC). In the revised Background we will expand this discussion and cite the 2023 IEEE ITSC study suggested by the reviewer on robust​ adaptive cruise control that explicitly accounts for tyre-road friction. We will clarify how our load estimates can serve as inputs to friction estimators and friction-adaptive controllers in ITS environments.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed the questions I raised.

Author Response

Thank you for pointing out the English and format problems in this second round, in addition to the formal and methodological ones already changed after the first round of review. A formal English Check my a native American speaker has been carried out. The figures have also been updated and waiting for the final look when fit into the PDF created by the Editor.

Reviewer 3 Report

Comments and Suggestions for Authors

   The author has addressed my concerns well. I would recommend the publication of this paper. However, the author can add more descriptions to highlight the core innovation of this work. Furthermore, the effect of the vertical load on the vehicle control strategy can refer to the recent work “A Direct Yaw Moment Control Framework Through Robust T-S Fuzzy Approach Considering Vehicle Stability Margin, IEEE/ASME Transactions on Mechatronics, vol. 29, no. 1, pp. 166-178, Feb. 2024”.

Author Response

Thank you for your comments and suggestion. The English is being improved through a check done by an American English native speaker and the figures updated but waiting for the final Editors layout of the document.

In relation with the proposed reference, the Yaw is not considered in the model as indicated in the text. So, after reviewing the reference proposal it doesn't seem to fit into the analysis already done.

Thanks again for your interest.