Pith. sign in

REVIEW 4 major objections 4 minor 34 references

Direct And Inverse Dynamics Problems For A Three-wheel Mobile Robot With Two Drive Wheels

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Closed-form torque formulas let a two-wheel-drive robot follow any smooth path.

desk verdict The Lagrange model and inverse dynamics are coherent but standard, and the closed-loop stability claim is not established because heading is never fed back. read the letter →

arxiv 2411.15318 v1 pith:GZN255JH submitted 2024-11-22 cs.RO math.OC

Roman Voliansky This is my paper · ORCID
classification cs.ROmath.OC
keywords mobilerobotdifferentialdriveinversedynamicsdirectLagrangeequationstrajectorytrackingsliding-modetorquecontrolmathematicalmodel
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

The paper solves the direct and inverse dynamics problems for a three-wheel mobile robot with two driven wheels. Its central claim is that the inverse problem has a closed-form algebraic solution: given a desired trajectory, the required wheel torques are obtained directly from the desired velocities and accelerations using expressions (14)-(15). This matters because trajectory following becomes a calculation rather than a matter of heuristic tuning. With a fast sliding-mode torque loop inside each drive, the author argues, the motor dynamics can be omitted, and the resulting two-channel closed-loop control system tracks complex trajectories with asymptotic stability.

What carries the argument

The load-bearing machinery is the Lagrangian description of the platform combined with a time-scale separation in the drive. The Lagrangian, built from the kinetic energy of the platform and wheels with zero potential energy, yields the mechanical equations in (9): the pair $\dot x=V\cos\varphi$, $\dot y=V\sin\varphi$ gives the nonholonomic kinematics, and the remaining pair expresses linear and angular acceleration as linear combinations of the two drive torques. The inverse solution simply inverts these equations: (14) solves the kinematics for $\varphi$, $V$, and $\omega$ from the desired path, and (15) solves the torque equations for $M_1$ and $M_2$. The sliding-mode torque regulator is the enabling assumption that removes motor dynamics and makes this inversion valid.

What would settle it

Simulate the full model given by (9) together with motor equations (11)-(13) while commanding a trajectory whose accelerations approach the torque-loop bandwidth; if tracking error grows in that regime, the inertia-free assumption is violated. A simpler check is to measure whether the actual wheel torques match $M_1$ and $M_2$ from (15) during a high-frequency maneuver.

Watch

Extended reading notes

Core claim

The discovery is that expressions (14)-(15) constitute the sought solution of the inverse dynamics problem for this robot. Equation (14) recovers the robot orientation and the linear and angular velocities of the platform from the desired Cartesian trajectory, while equation (15) gives the two drive torques $M_1$ and $M_2$ as linear combinations of the desired linear acceleration, the term $V\omega$, and the desired angular acceleration. Because a high-speed torque regulator running in sliding mode makes the torque loop effectively inertia-free, the motor equations can be set aside and the mechanical equations (9) are sufficient for the inversion. The resulting two-channel controller, closed by position feedback, reproduces the prescribed trajectory and is asymptotically stable, as shown in Fig. 4.

Load-bearing premise

The argument depends on each motor's torque loop being fast enough to be treated as instantaneous, so the torque commanded by the inverse solution appears at the wheels without motor dynamics.

Editorial extensions

If this is right

  • Trajectory tracking reduces to plugging the desired path and its derivatives into (14)-(15), so the nominal control law needs no iterative gain tuning.
  • Closing the position loop around the inverse controller corrects parameter mismatch and limits torque commands that would otherwise be physically unrealizable.
  • The inverse solution decouples the two drive channels: the left and right torque commands are separately determined by the desired motion.
  • For sufficiently smooth, low-frequency trajectories, the inertia-free torque-loop assumption holds and the closed-loop system should reproduce the path with asymptotic stability.

Reading between the lines

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

  • Editorial inference: modeling the torque loop as a first-order lag instead of ideal would add a lead term to (14)-(15), and the resulting predictions would show at what bandwidth tracking begins to degrade.
  • Editorial inference: the same algebraic inversion could serve as a feedforward block inside a model-predictive controller, shortening the prediction horizon needed for obstacle-avoidance maneuvers.
  • Editorial inference: parameterizing wheel slip in the kinematics would turn the single inverse solution into a family indexed by slip, giving a direct test of when the no-slip assumption breaks.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript (arXiv:2411.15318) develops a mathematical model of a three-wheel mobile robot with two driven wheels, using the Lagrange formalism to derive the direct dynamics (Eqs. (5)-(9)). It then solves the inverse dynamics problem: given a desired trajectory (X*, Y*), expressions (14)-(15) compute the required motor torques. A two-channel open-loop control structure is proposed, followed by a closed-loop version in Fig. 3 that adds position regulators on the Cartesian coordinates. The paper claims, based on the simulation in Fig. 4, that the closed-loop system exhibits high-quality trajectory tracking and asymptotic stability. The manuscript is written in Ukrainian with an English abstract.

Significance. The paper's main contribution is a clear, self-contained derivation of direct and inverse dynamics for a differential-drive robot with reluctance motors, and an explicit algebraic solution for feedforward torque computation. If validated, such expressions are useful for model-based control design. However, the claimed closed-loop stability is not rigorously supported, and the control architecture has a notable gap (no orientation feedback). The derivation itself is straightforward but presented transparently. The lack of a stability proof, controller parameters, or experimental validation limits the scientific impact; the work is more a modeling exercise than a complete control-systems contribution.

major comments (4)
  1. [Fig. 3] The closed-loop scheme in Fig. 3 feeds back only the Cartesian coordinates X and Y through the position regulators; the measured heading Ψ is displayed but not connected to any feedback path. For a nonholonomic differential-drive robot, the orientation is a state that determines the direction of the linear velocity. With no feedback of Ψ, an initial heading error or a heading disturbance will not be corrected, and asymptotic trajectory tracking is not guaranteed. The paper's claim of asymptotic stability after Fig. 4 is therefore not supported by the presented architecture. Please provide a stability analysis that includes the orientation state, or modify the scheme to feed back the full pose (or use an error transformation that explicitly depends on Ψ).
  2. [Text after Fig. 4] The assertion that 'the graphs show high quality reproduction of the desired trajectory and asymptotic stability of the synthesized control system' relies solely on visual inspection of a single simulation plot. No controller gains, robot parameters, initial conditions, or a mathematical definition of the desired trajectory are provided, and Fig. 4 has no axis labels or units. The result is therefore not reproducible. Provide a rigorous stability proof (e.g., a Lyapunov argument or a passivity analysis) and report all simulation parameters so that the claim can be independently verified.
  3. [Assumption before Eq. (14)] The inverse dynamics solution drops the motor dynamics (11)-(13) because a high-speed torque regulator with sliding modes is assumed to make the torque loop inertia-free. This is a strong assumption and is load-bearing: if the torque loop is not effectively instantaneous, the computed torques will not produce the desired trajectory. The paper does not justify this assumption with a time-scale separation argument or a robustness analysis. Please either justify the assumption quantitatively (e.g., by comparing the torque-loop bandwidth with the mechanical bandwidth) or include the motor dynamics in the simulation to demonstrate that the inverse-solution-based controller still performs well.
  4. [Eq. (14)] The inverse kinematic formula for the heading angle, Ψ = arctan(sy/sx), has a singularity when sx = 0 and is ambiguous by π unless the quadrant is resolved. For trajectories with points of zero velocity or with sign changes in the velocity components, this formula may produce discontinuous or incorrect heading commands. The paper should discuss these degenerate cases and propose a robust inversion (e.g., using atan2 with appropriate unwrapping) or state the class of trajectories for which the solution is valid.
minor comments (4)
  1. [Fig. 4] The figure lacks axis labels, units, and a legend; without these, the quality of tracking cannot be assessed quantitatively. Please add labeled axes, indicate the desired and actual trajectories, and specify the time scale.
  2. [Throughout] The manuscript is written in Ukrainian, whereas the abstract is in English. For a wider audience, consider providing a full English translation (or at least an extended English summary with the key equations and results).
  3. [References] The reference list contains many self-citations and some entries that are not clearly cited in the text. Please ensure that every reference is cited where relevant and that the list is trimmed to the most pertinent works.
  4. [Introduction] The introduction mentions methods based on fuzzy logic and machine learning as having subjectivity drawbacks; this is a broad claim without specific references. Either support the statement with concrete citations or soften the wording.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the inverse-dynamics result is a direct algebraic inversion of the plant equations, and the author's self-citations are not load-bearing.

full rationale

The paper builds a Lagrange-based model of the three-wheel robot, equations (5)-(9), and then obtains the inverse dynamics by solving the algebraic/differential equations of (9) for the required torques: equations (14)-(15) reconstruct linear and angular velocities from a desired Cartesian trajectory and then give M1 and M2 as explicit functions of those velocities. This is a standard computed-torque inversion: the output is what must be fed into the same equations to produce the desired motion, not a quantity that was assumed as an input. The inertia-free torque-loop assumption is stated explicitly before equation (14) as a design simplification, not imported as a conclusion. The closed-loop scheme in Fig. 3 and the simulation in Fig. 4 test the controller on the same model; this is self-consistent simulation rather than independent empirical validation, and the asymptotic-stability claim is not formally proved (and the heading is not fed back), but those are correctness and validation concerns, not circularity. The many cited works by the same author appear in the introductory survey and do not supply any load-bearing premise: equations (1)-(15) are fully derived in the paper without relying on an external uniqueness theorem or on a fitted parameter. No step reduces by construction or by self-citation to its own input, so no significant circularity is present.

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

The central claim rests on standard modeling assumptions: no slip, planar motion, center of mass at the axle midpoint, and an ideal torque loop via sliding modes. No numerical parameters are fitted to data. No new physical entities are introduced.

assumptions (6)
  • domain assumption The wheels roll without slipping
    Stated in the assumptions before Fig. 1; used to relate wheel angular velocities to platform linear and angular velocity in Eq. (9).
  • domain assumption The point Q at the middle of the drive axle coincides with the center of mass
    Stated in the assumptions; simplifies the kinetic energy in Eq. (5) by removing coupling terms between linear and angular motion.
  • domain assumption Motion is in a horizontal plane, so potential energy is zero
    Used after Eq. (3) to set the Lagrangian equal to the kinetic energy.
  • ad hoc to paper A high-speed torque regulator with sliding modes makes the motor dynamics inertia-free
    Introduced before Eq. (14) to justify solving the inverse problem using only Eq. (9) and dropping Eqs. (11)-(13). This is load-bearing for the inverse torque solution.
  • standard math Lagrange formalism applies to the electromechanical system
    Used in Eq. (1) to derive the mechanical equations of motion from kinetic energy.
  • domain assumption The switched reluctance motor is described by Eqs. (11)-(13)
    The per-phase voltage balance and electromagnetic torque equations are assumed to model the SRM drives.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Direct And Inverse Dynamics Problems For A Three-wheel Mobile Robot With Two Drive Wheels." pith.science (2026). https://pith.science/paper/GZN255JH

@misc{pith2026241115318,
  author       = {Pith},
  title        = {Pith review of: Direct And Inverse Dynamics Problems For A Three-wheel Mobile Robot With Two Drive Wheels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZN255JH}},
  note         = {Machine review of arXiv:2411.15318}
}
read the original abstract

Mobile robots are widely used to perform various technological operations in several sectors of the national economy. These operations are related to transporting goods and equipment, performing work to determine the condition of a technical object or structure, their construction or repair, performing work to study a specific territory and compile relevant maps, etc. Recently, the list of operations that mobile robots can perform has expanded with police and military operations. Obviously, the safety of personnel working nearby and the time required to perform the relevant operations depend on such robots' speed and accuracy of movement. Therefore, an important task arises to study and form the trajectories of movement of mobile robots. Optimization, adaptation, robustness methods, and the theory of movement stability allow us to consider a mobile robot as a dynamic system with several inputs and outputs. The mathematical description of such a dynamic system can be used to analyze and synthesize the desired trajectories of movement by solving the corresponding direct and inverse dynamics problems. Therefore, creating a mathematical model of a mobile robot is a relevant task, the solution of which allows us to create and research robot control systems that ensure movement along predetermined desired trajectories.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 19 canonical work pages

  1. [1]

    Таке може статися внаслідок зміни умов роботи, впливу оточуючого середовища, тощо

    Вона є розімкненою відносно координат робота, що може призвести до неточного відпр ацювання бажано ї траєкторії у в ипадку коли параметри, використовувані при налаштуванні блоку формувача моменту відрізняються від реальних параметрів робот а. Таке може статися внаслідок зміни умов роботи, впливу оточуючого середовища, тощо

    work page

  2. [2]

    Задаючи м оменти, які формуються в цій структ урі можуть досягати великих значень, які неможна відтворит и фізично , внаслідок невірного завдання бажаної траєкторії руху. Колісний робот Двигун 1 Двигун 2 Регулятор моменту 1 Регулятор моменту 2 Датчик положення ротора Датчик положення ротора Формувач моментів X* Y* M1* M2* U1 U2 θ1 θ2 X Y Ψ Рисунок 2 – Фун...

    work page

  3. [3]

    Ackerman M obile Robot with Arm,

    R. A. Orozco -Velázquez et al., "Ackerman M obile Robot with Arm," 2016 International Conference on Mechatronics, Electronics and Automotive Engineering (ICMEAE), Cuernavaca, 2016, pp. 55-60

    work page 2016

  4. [4]

    R -Crank: Amphibious all terrain mobile robot,

    S. Yamada, S. Hirose, G. En do, K. Suzumori and H . Nabae, "R -Crank: Amphibious all terrain mobile robot," 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, 2016, pp. 1067-1072

    work page 2016

  5. [5]

    Distance contro lled rescue and security mobile robot,

    G. K. Dey, R. Hossen, M. S. Noor and K. T. Ahmmed, "Distance contro lled rescue and security mobile robot," 2013 International Conference on In formatics, Electronics and Vision (ICIEV), Dhaka, 2013, pp. 1-6

    work page 2013

  6. [6]

    Strategy for designing a control system for a target-approach task by a mobil e robot,

    M. Przybyła, K. Łakomy and P. Herman, "Strategy for designing a control system for a target-approach task by a mobil e robot," 2014 IEEE/A SME 10th International Conference on Mechatronic and Embedded Systems and Applications (MESA), Senigallia, 2014, pp. 1 - 6

    work page 2014

  7. [7]

    Inverse kinematics of redundant robots using genetic algorithms,

    J. K. Par ker, A. R. Khoogar and D. E. Goldberg, "Inverse kinematics of redundant robots using genetic algorithms, " Proceedings, 1989 I nternational Conference on Robotics and Automation, Scottsdale, AZ, 1989, pp. 271-276 vol.1

    work page 1989

  8. [8]

    Control and stabilization of nonholonomic dynamic systems,

    M. Bloch, M. Reyhanoglu, and N. H. McClamroch, "Control and stabilization of nonholonomic dynamic systems," IEEE Transactions on Automatic Con trol, vol. 37, no. 11, pp. 1746-1757, Nov. 1992

    work page 1992

Show all 34 references
  1. [9]

    Control of nonholonomic systems using reference vector fields,

    D. Panagou, H. G. Tanner and K. J. Kyriakopoulos, "Control of nonholonomic systems using reference vector fields," 2011 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, 2011, pp. 2831-2836

    2011

  2. [10]

    Model Predictive Control for the navigation of a nonholonomic vehicle with field -of-view constraints,

    S. Maniatopoulos, D. Panagou and K. J. Kyriakopoulos, "Model Predictive Control for the navigation of a nonholonomic vehicle with field -of-view constraints," 2013 American Control Conference, Washington, DC, 2013, pp. 3967-3972

    2013

  3. [11]

    Fast mo tion planning for multiple moving ro bots,

    S. J. Buckley, "Fast mo tion planning for multiple moving ro bots," Proceedings, 1989 International Conference on Robotics and Automation, Scottsdale, AZ, 1989, pp. 322 -326 vol.1

    1989

  4. [12]

    Moving target tracking by mobile robot,

    P. Anjaly, A. Devanand and V. R. Jisha, "Moving target tracking by mobile robot, " 2014 International Conference on Power Signals Cont rol and Computations (EPSCICON), Thrissur, 2014, pp. 1-6

    2014

  5. [13]

    Синтез оптимально ї системи керування з іраціональною активаційною функцією // Вісник НТУ «ХПІ» «Проблеми автоматизованого електроприводу Теорія та практика)»

    Садовой O.В. Синтез оптимально ї системи керування з іраціональною активаційною функцією // Вісник НТУ «ХПІ» «Проблеми автоматизованого електроприводу Теорія та практика)». – Харків: НТУ «ХПИ». – 2010.– Вип. 28. – С.49 – 51 (на рос.)

    2010

  6. [14]

    Синтез оптимально ї системи керування з нел інійною активаційною функцією

    Садовой О.В. Синтез оптимально ї системи керування з нел інійною активаційною функцією. Електротехнічні і комп'ютерні системи. Київ: Техніка. 2014, № 15 (91), С. 69 – 71 (на рос.)

    2014

  7. [15]

    Математич ні моделі усунення помпажа в відцентровому компресорі // Вісник КДУ ім

    Цaбенко М.В. Математич ні моделі усунення помпажа в відцентровому компресорі // Вісник КДУ ім. М. Остоградського. - 2010. - Випуск 4(63). - С. 167 - 169. (на рос.)

    2010

  8. [16]

    Система оптимального керування натягом прокатуваної полос и стану холодної прокатки // З бірка наукових прац ь Донбаського державного техн ічного університету

    Назарова Е.С. Система оптимального керування натягом прокатуваної полос и стану холодної прокатки // З бірка наукових прац ь Донбаського державного техн ічного університету. –2011. – Вип. 34. – С.122-130 (на рос.)

    2011

  9. [17]

    Система керування сл ідкуючим електроприводом із ковзним режимом 2 -го порядка, Вісник Кременчуцького державного університету імені Михайла Остроградського

    Волянський Р.С. Система керування сл ідкуючим електроприводом із ковзним режимом 2 -го порядка, Вісник Кременчуцького державного університету імені Михайла Остроградського. – Кременчук: КДУ, 2010. – Вип. 4/2010 (63) частина 3. – с.11-14 (на рос.)

    2010

  10. [18]

    In: IEEE Int ernational Conference on Modern Electrical and Energy Systems (MEES), Kremenchuk, Ukraine, pp

    Sokhina, Y Active Suspension Control System. In: IEEE Int ernational Conference on Modern Electrical and Energy Systems (MEES), Kremenchuk, Ukraine, pp. 10-13 (2019)

    2019

  11. [19]

    Волянський Р. С. Синтез керу ючого впливу для електромеханічних об ’єктів з невідомими параметрами // Вісник НТУ «ХПІ». — Харьков: ХПИ. — 2015. — № 12 (1121). — С. 60—63 (на рос.)

    2015

  12. [20]

    Pranolo А, Parallel Mathematical Models of Dynamic Objects, Int. J. Adv. Intell. Informs. 4(2) (2018) 120–131

    2018

  13. [21]

    Anti -swing Control System for the One Class of Underactuated Dynamic Objects,

    I. Shramko, "Anti -swing Control System for the One Class of Underactuated Dynamic Objects," 2020 IEEE Problems of Automated Electrodrive. Theory and Practice (PAEP), Kremenchuk, Ukraine, 2020, pp. 1-4

    2020

  14. [22]

    Chaotic Time -variant Dynamical System,

    Kluev, O., "Chaotic Time -variant Dynamical System," 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), 2020, pp. 606-609

    2020

  15. [23]

    Chua's circuits interval synchronization,

    R. Voliansky, "Chua's circuits interval synchronization," 2017 4th International Scientific - Practical Conference Problems of Infocommunications. Science and Technology (PIC S&T), Kharkov, Ukraine, 2017, pp. 439-443

    2017

  16. [24]

    Sliding Mode Control for DC Generator with Uncertain Load,

    O. Sinkevych, "Sliding Mode Control for DC Generator with Uncertain Load," 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), Lviv -Slavske, Ukraine, 2020, pp. 313-316

    2020

  17. [25]

    Voliansky, R. (2017). Transformation of the generalized chaotic system into canonical form. International Journal of Advances in Intelligent Informatics, 3(3), 117–124

    2017

  18. [26]

    The transformation of linear dynamical object's equation to Brunovsky canonical form,

    Sadovoi A. V., "The transformation of linear dynamical object's equation to Brunovsky canonical form," 2017 IEEE 4th International Conference Actual Problems of Unmanned Aerial Vehicles Developments (APUAVD), Kiev, Ukraine, 2017, pp. 196-199

    2017

  19. [27]

    Садовой А. В. Дослідження частотн их характеристик динам ічних ланок із похідним дробових порядків / Р. С. Волянский, // Збірка наукових праць ДДТУ (технічні науки). Тематичний випуск „Проблеми автоматизованого електроприводу. Теорія і практика”. – 2007. – С. 82–85. (на рос.)

    2007

  20. [28]

    Construction of Parallel Piecewise -Linear Interval Models for Nonlinear Dynamical Objects,

    O. Sinkevych, "Construction of Parallel Piecewise -Linear Interval Models for Nonlinear Dynamical Objects," 2019 9th International Conference on Advanced Computer Information Technologies (ACIT), 2019, pp. 97-100

    2019

  21. [29]

    Sliding Mode Interval Controll er for the Mobile Robot,

    Y. Sokhina, "Sliding Mode Interval Controll er for the Mobile Robot," 2019 XIth International Scientific and Practical Conference on Electronics and Information Technologies (ELIT), Lviv, Ukraine, 2019, pp. 76-81

    2019

  22. [30]

    Interval model of the piezoelectric drive,

    N. Volianska, "Interval model of the piezoelectric drive," 2018 14th International Conference on Advanced Trends in Radioelecrtronics, Telecommunications and Computer Engineering (TCSET), Lviv-Slavske, Ukraine, 2018, pp. 1-6

    2018

  23. [31]

    Root Methods for Dynamic Analysis of the One Class Chaotic Systems,

    O. Sinkevych, "Root Methods for Dynamic Analysis of the One Class Chaotic Systems," 2019 IEEE 14th International Con ference on Computer Sciences and Information Technologies (CSIT), 2019, pp. 117-121

    2019

  24. [32]

    Defining of Lyapunov Functions for the Generalized Nonlinear Object,

    O. Sadovoi, "Defining of Lyapunov Functions for the Generalized Nonlinear Object," 2018 IEEE 5th International Conference on Methods and Systems of Navigation and Motion Control (MSNMC), Kiev, Ukraine, 2018, pp. 222-228

    2018

  25. [33]

    ХПІ": зб. науч. пр. Темат. в ип. : Механ іко-технологічні системи та комплекси. – Харків: НТУ

    Садовой О. В. Лінеаризація зворотними зв’язками рівнянь динаміки узагальненого електромеханічного об’єкта з диференціальним рівнянням спостережуваності // Вісник Нац. техн. ун-ту "ХПІ": зб. науч. пр. Темат. в ип. : Механ іко-технологічні системи та комплекси. – Харків: НТУ "ХП...

    2014

  26. [34]

    Electromechanical System Motion Control in Direct and Inverse Time,

    N. Krasnoshapka, O. Statsenko, "Electromechanical System Motion Control in Direct and Inverse Time," 2022 IEEE 3rd KhPI Week on Advanced Technology (KhPIWeek), Kharkiv, Ukraine, 2022, pp. 1-6

    2022

Pith tools

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