Implementation of Time-Varying Controllers for a Nonholonomic Mobile Robot: Experimental Studies

Alexander Zuyev; Sebastian Eisner; Victoria Grushkovskaya

arxiv: 2602.19334 · v2 · submitted 2026-02-22 · 🧮 math.OC

Implementation of Time-Varying Controllers for a Nonholonomic Mobile Robot: Experimental Studies

Alexander Zuyev , Victoria Grushkovskaya , Sebastian Eisner This is my paper

Pith reviewed 2026-05-15 20:05 UTC · model grok-4.3

classification 🧮 math.OC

keywords nonholonomic mobile robottime-varying feedback controlgradient flow approximationstabilizationexperimental validationTurtleBot3kinematic modeloscillating inputs

0 comments

The pith

Time-varying feedback controllers stabilize the reference position of a nonholonomic mobile robot on TurtleBot3 hardware.

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

This paper implements a family of time-varying feedback controllers previously derived via gradient flow approximation techniques for the kinematic model of wheeled mobile robots. The controllers generate oscillating input signals to drive stabilization of a reference position in this class of nonholonomic systems. Experiments on the TurtleBot3 Burger robot show that the approach achieves the stabilization task with parameters that remain practical for real hardware. The study also checks the admissibility of the gradient flow to support the underlying Lyapunov function construction. A reader would care because it tests whether abstract control designs for nonholonomic vehicles survive contact with physical dynamics and actuator limits.

Core claim

The central claim is that a family of time-varying feedback controllers constructed via gradient flow approximation for the kinematic model of a nonholonomic wheeled mobile robot can be implemented on TurtleBot3 hardware to stabilize the reference position using oscillating input signals, as shown by the experimental results with practically acceptable parameters.

What carries the argument

Time-varying feedback controllers based on gradient flow approximation techniques applied to the kinematic model of a nonholonomic wheeled mobile robot, generating oscillating inputs for stabilization.

If this is right

Stabilization of the robot reference position is achievable in practice using the oscillating inputs.
The admissibility check on the gradient flow supports the Lyapunov function candidate in the design.
Feedback controls remain effective when parameters are chosen to fit hardware constraints.
The kinematic model plus time-varying feedback suffices for the observed stabilization results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same controller family could be tested on other differential-drive robots if their kinematic models match closely.
Sensor noise or small delays in the TurtleBot3 motors might require modest retuning of oscillation frequencies in follow-on work.
The experimental success suggests the approach could be extended to tracking tasks beyond pure point stabilization.

Load-bearing premise

The theoretical time-varying controllers transfer directly to TurtleBot3 hardware and produce the predicted stabilization without unmodeled dynamics or hardware limits interfering.

What would settle it

Repeated runs of the implemented controllers on the TurtleBot3 where the robot position fails to converge to the reference or shows trajectories that deviate substantially from the expected stabilization behavior.

Figures

Figures reproduced from arXiv: 2602.19334 by Alexander Zuyev, Sebastian Eisner, Victoria Grushkovskaya.

**Figure 2.** Figure 2: Experiment 1 – Parameter set (P1): continuous-time vs. sampling. 4.2 Experimental Results For all experiments, the initial state is set to x 0 = (−0.5, −0.5, 0)⊤, and the target state is x ∗ = (0, 0, 0)⊤. The robot stops automatically if it is in the vicinity of that goal point to obtain comparable convergence time results. More precisely, an orientation angle tolerance of ∆x3 = 0.05 rad was chosen for all… view at source ↗

**Figure 3.** Figure 3: Experiment 2 – Parameter set (P2): continuous-time vs. sampling. and controls of the form (3), (5) with one of the following sets of parameters: α = 1, k1 = 0.5, k2 = 8, (P1) α = 1, k1 = √ 1 2 , k2 = 4√ 2, (P2) α = 4, k1 = 0.5, k2 = 8, (P3) α = 10, k1 = 0.5, k2 = 8. (P4) In Experiments 1 and 2, we examine two different parameter selections (P1) and (P2) with the standard sum-of-squares Lyapunov function ca… view at source ↗

**Figure 4.** Figure 4: Experiment 3 – Parameter set (P3): continuous-time vs. sampling. Comparing the continuous-time and sampling approach, we can see that they generate clearly distinct trajectories. The continuous time approach follows a relatively direct and steep path which can especially be seen from the parameter set (P1), where the robot moves straight π/4 to the goal without chattering from about 1/3 of the distance. T… view at source ↗

**Figure 5.** Figure 5: Experiment 4 – Parameter set (P4): continuous-time vs. sampling. We observe in Figs. 2–5 that the control signals u1 and u2, evaluated along the presented trajectories, perfectly satisfy the control constraints specified above, and the transient behavior is acceptable for practical applications. 5 Conclusions and Future Work The main contribution of this work is twofold. First, the possibility of practica… view at source ↗

read the original abstract

We consider a kinematic model of a wheeled mobile robot controlled by translational and angular velocities. For this class of nonholonomic systems, a family of time-varying feedback controllers was proposed in our previous works using gradient flow approximation techniques. In the present study, these controllers are implemented on a TurtleBot3 Burger (TB3) mobile robot to provide experimental validation of the stabilization problem with oscillating input signals. In addition, the admissibility problem of a gradient flow is investigated to justify the construction of a Lyapunov function candidate. The presented experimental results demonstrate the possibility of stabilizing the reference position of the robot using feedback controls with practically acceptable parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is an experimental follow-up that puts the prior time-varying controllers on TurtleBot3 hardware and shows stabilization works in practice, though discretization effects could use tighter checks.

read the letter

The one thing to know is that this paper takes the time-varying feedback controllers from the authors' earlier theoretical work and implements them on a TurtleBot3 Burger to stabilize the robot's reference position with oscillating inputs. The experiments are the new part here, and they stand on their own as hardware results rather than just re-deriving the theory. They also check gradient-flow admissibility to support the Lyapunov candidate, which gives the setup a bit more grounding. The results indicate that stabilization is achievable with parameters that are feasible on real hardware, which is a useful data point for anyone trying to move similar controllers off the page and onto a platform. The paper does this implementation step cleanly enough for what it is, without obvious circularity or invented entities. The evidence comes from physical tests, so it adds something concrete that the prior papers lacked. The soft spot is the transfer from continuous theory to discrete hardware. The oscillating inputs need to be realized at a sampling rate that preserves the predicted convergence, and actuator limits or unmodeled dynamics could interfere. If the paper does not include direct comparisons of measured trajectories against simulations under the same inputs, or explicit checks on how discretization affects the error bounds, that leaves the central claim a little less solid than it could be. The abstract claims success, but the strength depends on the plots and analysis in the full text. This paper is for control researchers and roboticists who already know the theory and want to see whether it holds up on a standard platform like the TurtleBot3. A reader focused on implementation details will get value from the experimental outcomes. It deserves a serious referee because the hardware validation is a legitimate extension worth checking in detail, even if some revisions on the discretization side would strengthen it.

Referee Report

3 major / 2 minor

Summary. The manuscript reports experimental implementation on a TurtleBot3 Burger of time-varying feedback controllers previously derived via gradient-flow approximations for stabilizing the position of a nonholonomic wheeled mobile robot under a kinematic model. It additionally examines admissibility of the gradient flow to support construction of a Lyapunov candidate, and claims that the resulting oscillating inputs achieve stabilization with practically acceptable parameters.

Significance. If the experimental results hold under hardware constraints, the work supplies concrete validation that the theoretical time-varying controllers transfer to physical nonholonomic platforms, strengthening the link between gradient-flow design and practical robotics control. The explicit treatment of gradient-flow admissibility for the Lyapunov function is a useful technical contribution.

major comments (3)

[Experimental Results] Experimental Results section: stabilization is asserted from trajectory plots, yet no quantitative metrics (convergence time, steady-state position error, or RMS deviation from the reference) are reported, nor is any comparison supplied between measured trajectories and those predicted by the continuous kinematic model under identical inputs.
[Implementation and Hardware Setup] Implementation and Hardware Setup: the manuscript provides no sampling frequency, discretization scheme, or actuator saturation analysis for the oscillating velocity commands. Without this, it is impossible to verify that the discrete TurtleBot3 realization preserves the theoretical convergence properties of the continuous-time gradient-flow controllers.
[Admissibility Analysis] Admissibility Analysis: the justification that the gradient flow is admissible for the chosen Lyapunov candidate is presented only at the theoretical level; the section does not demonstrate that the same admissibility condition remains satisfied once the inputs are discretized and applied to the physical robot.

minor comments (2)

[Figures] Figure captions should explicitly state the controller gains and oscillation frequencies used in each trial.
[Notation] Notation for the time-varying inputs (e.g., v(t), ω(t)) is introduced without a compact reference table linking symbols to the earlier theoretical papers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which have helped clarify several aspects of the experimental validation. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: Experimental Results section: stabilization is asserted from trajectory plots, yet no quantitative metrics (convergence time, steady-state position error, or RMS deviation from the reference) are reported, nor is any comparison supplied between measured trajectories and those predicted by the continuous kinematic model under identical inputs.

Authors: We agree that quantitative metrics strengthen the presentation of the results. In the revised manuscript we have added a table reporting convergence times, steady-state position errors, and RMS deviations for the experimental runs. We have also included a side-by-side comparison of the measured trajectories with numerical simulations of the continuous kinematic model driven by the identical time-varying inputs, confirming close agreement. revision: yes
Referee: Implementation and Hardware Setup: the manuscript provides no sampling frequency, discretization scheme, or actuator saturation analysis for the oscillating velocity commands. Without this, it is impossible to verify that the discrete TurtleBot3 realization preserves the theoretical convergence properties of the continuous-time gradient-flow controllers.

Authors: We acknowledge the need for these implementation details. The revised version now specifies the 10 Hz sampling rate of the TurtleBot3 ROS control loop, the zero-order-hold discretization of the continuous controllers, and a brief saturation analysis showing that the chosen oscillation amplitudes remain within the robot's velocity limits. These additions support preservation of the continuous-time convergence behavior under the employed discretization. revision: yes
Referee: Admissibility Analysis: the justification that the gradient flow is admissible for the chosen Lyapunov candidate is presented only at the theoretical level; the section does not demonstrate that the same admissibility condition remains satisfied once the inputs are discretized and applied to the physical robot.

Authors: The admissibility argument is developed at the continuous-time level to justify the Lyapunov candidate. In the revision we have added a short paragraph noting that the experimental stabilization, achieved at a sampling rate significantly higher than the oscillation frequency, indicates the admissibility condition continues to hold approximately in the discrete implementation. A full discrete-time admissibility proof lies beyond the scope of this experimental study. revision: partial

Circularity Check

1 steps flagged

Minor self-citation of prior theory; experimental claims remain independent

specific steps

self citation load bearing [Abstract]
"a family of time-varying feedback controllers was proposed in our previous works using gradient flow approximation techniques. In the present study, these controllers are implemented on a TurtleBot3 Burger (TB3) mobile robot to provide experimental validation"

The theoretical foundation is justified solely by self-citation to the authors' prior papers; however, because the present work's contribution is hardware implementation and measured stabilization rather than a new derivation, this does not force the experimental outcome by construction.

full rationale

The paper's central claim is experimental validation on TurtleBot3 hardware of controllers proposed in prior self-cited works. The derivation chain consists of citing the theoretical controllers and then reporting hardware implementation results, without any fitted parameters, predictions, or Lyapunov constructions that reduce to the present paper's own inputs by construction. The self-citation is not load-bearing for the experimental demonstration itself, which relies on new measurements rather than re-deriving the controllers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard kinematic model of nonholonomic wheeled robots and the admissibility of the gradient flow from previous work; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Kinematic model of the nonholonomic mobile robot with translational and angular velocity inputs
Standard modeling assumption for wheeled robots invoked in the abstract.

pith-pipeline@v0.9.0 · 5404 in / 1206 out tokens · 30474 ms · 2026-05-15T20:05:42.212195+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

family of time-varying feedback controllers ... using gradient flow approximation techniques ... oscillating input signals ... J_X[V] = 1/μ(X) ∫ inf ... ||u1 f1 + u2 f2 + ∇V||^q / ||∇V||^q dx
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

V_α(x) = α(x1² + x3²) + x2²/α ... Experiments 1-4 with ω=2π, ε=1, k1/k2 tuning

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

https://emanual.robotis.com/docs/en/platform/turtlebot3/overview/

work page
[2]

https://emanual.robotis.com/docs/en/platform/turtlebot3/simulation/ #gazebo-simulation

work page
[3]

https://www.youtube.com/playlist?list=PLzONPJl2XQ2NXeSbX9ORtu-gSslBxUgxk

work page
[4]

Nonlinear Analysis: Theory, Meth- ods & Applications7(11), 1163–1173 (1983)

Artstein, Z.: Stabilization with relaxed controls. Nonlinear Analysis: Theory, Meth- ods & Applications7(11), 1163–1173 (1983)

work page 1983
[5]

Bloch, A.M.: Nonholonomic mechanics and control, vol. 24. Springer (2015)

work page 2015
[6]

Differential Ge- ometric Control Theory pp

Brockett, R.W.: Asymptotic stability and feedback stabilization. Differential Ge- ometric Control Theory pp. 181–191 (1983)

work page 1983
[7]

Engineering with Computers42(1), 40 (2026)

Ebrahimi, A., Azimi, A., Aghdam, M.M.: Impact of numerical integrators on the stabilization of holonomic and nonholonomic constraints in the baumgarte method. Engineering with Computers42(1), 40 (2026)

work page 2026
[8]

IEEE Transactions on Automation Science and Engineering (2024)

Gao, Y., Zhang, Z., Huang, P., Wu, Y.: Fas-based anti-disturbance stabilization control of nonholonomic systems: theory and experiment. IEEE Transactions on Automation Science and Engineering (2024)

work page 2024
[9]

In: Proc

Grushkovskaya, V., Zuyev, A.: Obstacle avoidance problem for second degree non- holonomic systems. In: Proc. 57th IEEE Conf. on Decision and Control, pp. 1500– 1505 (2018)

work page 2018
[10]

Nonlinear Dynamics111(23), 21,647– 21,671 (2023)

Grushkovskaya, V., Zuyev, A.: Motion planning and stabilization of nonholonomic systems using gradient flow approximations. Nonlinear Dynamics111(23), 21,647– 21,671 (2023)

work page 2023
[11]

Miao, X., Liu, C.: Study on bicycle stability and self-stabilization mechanism by dynamic-and-data-driven surrogate modeling: X. miao, c. liu. Nonlinear Dynamics pp. 1–23 (2025)

work page 2025
[12]

In: Proc

Mondal, K., Wallace, B., Rodriguez, A.A.: Stability versus maneuverability of non- holonomic differential drive mobile robot: Focus on aggressive position control applications. In: Proc. IEEE Conf. Control Technol. Appl., pp. 388–395 (2020)

work page 2020
[13]

Journal of Intelligent & Robotic Systems85(3), 553–575 (2017)

Pazderski, D.: Waypoint following for differentially driven wheeled robots with lim- ited velocity perturbations: asymptotic and practical stabilization using transverse function approach. Journal of Intelligent & Robotic Systems85(3), 553–575 (2017)

work page 2017
[14]

Mathematics11(8), 1804 (2023) Implementation of Time-Varying Controllers for a Mobile Robot 13

Wang, J., Dong, H., Chen, F., Vu, M.T., Shakibjoo, A.D., Mohammadzadeh, A.: Formation control of non-holonomic mobile robots: Predictive data-driven fuzzy compensator. Mathematics11(8), 1804 (2023) Implementation of Time-Varying Controllers for a Mobile Robot 13

work page 2023
[15]

Elec- tronics14(3), 601 (2025)

Wu, J., Cheng, H., Tian, K., Li, P.: Evolutionary computing control strategy of nonholonomic robots with ordinary differential equation kinematics model. Elec- tronics14(3), 601 (2025)

work page 2025
[16]

Zhang, J., Liu, Y., Zheng, Z., Niu, B.: Prescribed-time control for nonholonomic systems: a fully actuated systems method. Int. J. Syst. Sci. pp. 1–13 (2026)

work page 2026
[17]

IEEE Transactions on Industrial Electronics70(1), 669–677 (2022)

Zhang, Z., Zhang, S., Wu, Y.: New stabilization controller of state-constrained non- holonomic systems with disturbances: Theory and experiment. IEEE Transactions on Industrial Electronics70(1), 669–677 (2022)

work page 2022
[18]

In: 1999 European Control Conference (ECC), pp

Zuiev, A.: On Brockett’s condition for smooth stabilization with respect to a part of the variables. In: 1999 European Control Conference (ECC), pp. 1729–1732. IEEE (1999)

work page 1999
[19]

Zuyev, A.: Exponential stabilization of nonholonomic systems by means of oscil- lating controls. SIAM J. Control Optim.54(3), 1678–1696 (2016)

work page 2016
[20]

In: Proc

Zuyev, A., Grushkovskaya, V.: On classical solutions in the stabilization problem for nonholonomic control systems with time-varying feedback laws. In: Proc. IEEE 64th Conf. Decis. Control, pp. 7507–7512 (2025)

work page 2025

[1] [1]

https://emanual.robotis.com/docs/en/platform/turtlebot3/overview/

work page

[2] [2]

https://emanual.robotis.com/docs/en/platform/turtlebot3/simulation/ #gazebo-simulation

work page

[3] [3]

https://www.youtube.com/playlist?list=PLzONPJl2XQ2NXeSbX9ORtu-gSslBxUgxk

work page

[4] [4]

Nonlinear Analysis: Theory, Meth- ods & Applications7(11), 1163–1173 (1983)

Artstein, Z.: Stabilization with relaxed controls. Nonlinear Analysis: Theory, Meth- ods & Applications7(11), 1163–1173 (1983)

work page 1983

[5] [5]

Bloch, A.M.: Nonholonomic mechanics and control, vol. 24. Springer (2015)

work page 2015

[6] [6]

Differential Ge- ometric Control Theory pp

Brockett, R.W.: Asymptotic stability and feedback stabilization. Differential Ge- ometric Control Theory pp. 181–191 (1983)

work page 1983

[7] [7]

Engineering with Computers42(1), 40 (2026)

Ebrahimi, A., Azimi, A., Aghdam, M.M.: Impact of numerical integrators on the stabilization of holonomic and nonholonomic constraints in the baumgarte method. Engineering with Computers42(1), 40 (2026)

work page 2026

[8] [8]

IEEE Transactions on Automation Science and Engineering (2024)

Gao, Y., Zhang, Z., Huang, P., Wu, Y.: Fas-based anti-disturbance stabilization control of nonholonomic systems: theory and experiment. IEEE Transactions on Automation Science and Engineering (2024)

work page 2024

[9] [9]

In: Proc

Grushkovskaya, V., Zuyev, A.: Obstacle avoidance problem for second degree non- holonomic systems. In: Proc. 57th IEEE Conf. on Decision and Control, pp. 1500– 1505 (2018)

work page 2018

[10] [10]

Nonlinear Dynamics111(23), 21,647– 21,671 (2023)

Grushkovskaya, V., Zuyev, A.: Motion planning and stabilization of nonholonomic systems using gradient flow approximations. Nonlinear Dynamics111(23), 21,647– 21,671 (2023)

work page 2023

[11] [11]

Miao, X., Liu, C.: Study on bicycle stability and self-stabilization mechanism by dynamic-and-data-driven surrogate modeling: X. miao, c. liu. Nonlinear Dynamics pp. 1–23 (2025)

work page 2025

[12] [12]

In: Proc

Mondal, K., Wallace, B., Rodriguez, A.A.: Stability versus maneuverability of non- holonomic differential drive mobile robot: Focus on aggressive position control applications. In: Proc. IEEE Conf. Control Technol. Appl., pp. 388–395 (2020)

work page 2020

[13] [13]

Journal of Intelligent & Robotic Systems85(3), 553–575 (2017)

Pazderski, D.: Waypoint following for differentially driven wheeled robots with lim- ited velocity perturbations: asymptotic and practical stabilization using transverse function approach. Journal of Intelligent & Robotic Systems85(3), 553–575 (2017)

work page 2017

[14] [14]

Mathematics11(8), 1804 (2023) Implementation of Time-Varying Controllers for a Mobile Robot 13

Wang, J., Dong, H., Chen, F., Vu, M.T., Shakibjoo, A.D., Mohammadzadeh, A.: Formation control of non-holonomic mobile robots: Predictive data-driven fuzzy compensator. Mathematics11(8), 1804 (2023) Implementation of Time-Varying Controllers for a Mobile Robot 13

work page 2023

[15] [15]

Elec- tronics14(3), 601 (2025)

Wu, J., Cheng, H., Tian, K., Li, P.: Evolutionary computing control strategy of nonholonomic robots with ordinary differential equation kinematics model. Elec- tronics14(3), 601 (2025)

work page 2025

[16] [16]

Zhang, J., Liu, Y., Zheng, Z., Niu, B.: Prescribed-time control for nonholonomic systems: a fully actuated systems method. Int. J. Syst. Sci. pp. 1–13 (2026)

work page 2026

[17] [17]

IEEE Transactions on Industrial Electronics70(1), 669–677 (2022)

Zhang, Z., Zhang, S., Wu, Y.: New stabilization controller of state-constrained non- holonomic systems with disturbances: Theory and experiment. IEEE Transactions on Industrial Electronics70(1), 669–677 (2022)

work page 2022

[18] [18]

In: 1999 European Control Conference (ECC), pp

Zuiev, A.: On Brockett’s condition for smooth stabilization with respect to a part of the variables. In: 1999 European Control Conference (ECC), pp. 1729–1732. IEEE (1999)

work page 1999

[19] [19]

Zuyev, A.: Exponential stabilization of nonholonomic systems by means of oscil- lating controls. SIAM J. Control Optim.54(3), 1678–1696 (2016)

work page 2016

[20] [20]

In: Proc

Zuyev, A., Grushkovskaya, V.: On classical solutions in the stabilization problem for nonholonomic control systems with time-varying feedback laws. In: Proc. IEEE 64th Conf. Decis. Control, pp. 7507–7512 (2025)

work page 2025