Rapid Vibration Suppression and Trajectory Tracking of a Serial Manipulator with Multi-Flexible Links

Chengyi Wang; Ji Wang; Yilong Huang

arxiv: 2605.17477 · v1 · pith:ITKWC4Q6new · submitted 2026-05-17 · 💻 cs.RO

Rapid Vibration Suppression and Trajectory Tracking of a Serial Manipulator with Multi-Flexible Links

Chengyi Wang , Yilong Huang , Ji Wang This is my paper

Pith reviewed 2026-05-20 12:40 UTC · model grok-4.3

classification 💻 cs.RO

keywords serial flexible manipulatorsvibration suppressionbackstepping boundary controlTimoshenko beam modelDeepONet approximationtrajectory trackingoutput feedback

0 comments

The pith

A backstepping boundary controller with DeepONet kernel approximation rapidly suppresses vibrations and tracks trajectories in multi-link flexible serial manipulators using only boundary measurements.

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

The paper addresses vibrations in lightweight multi-link robotic arms that limit their speed and accuracy. It transforms the system model into hyperbolic partial differential equations and designs a boundary controller via backstepping to add damping along each link. This enables quick vibration reduction and precise tip tracking based solely on measurements at the joints. A neural operator approximates the control kernels to make the method computationally feasible in real time. Experimental results on a two-link setup confirm superior performance over linear quadratic regulator control.

Core claim

Each link-joint is modeled as a Timoshenko beam coupled with an ODE and transformed into a canonical hyperbolic PDE with boundary dynamics. A backstepping-based boundary controller at the joint is developed to equivalently inject distributed damping along the beam, enabling rapid vibration suppression and trajectory tracking, only using available boundary measurements, with a DeepONet-based approximation for practical deployment.

What carries the argument

Backstepping-based boundary controller at the joint that equivalently injects distributed damping into the hyperbolic PDE model of the flexible links.

If this is right

Vibration suppression occurs faster than with LQR feedforward control.
The end-effector converges to the desired trajectory more reliably.
The approach scales to n-link manipulators with lower computational cost via DeepONet.
Controller updates remain feasible under varying operating conditions.

Where Pith is reading between the lines

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

This strategy may support higher operational speeds in flexible robots without added fatigue.
The PDE modeling and boundary control could extend to other distributed flexible systems like antennas or cranes.
Neural operator approximation of kernels might enable adaptive versions for changing loads.

Load-bearing premise

The Timoshenko beam model with ODE coupling and transformation to canonical hyperbolic PDE accurately represents the dynamics of the multi-flexible-link serial manipulator.

What would settle it

A test on the physical two-link manipulator where vibration settling times or end-effector tracking errors show no improvement under the proposed controller compared to LQR with feedforward control.

read the original abstract

Flexible robotic manipulators (FRMs) offer advantages in lightweight design and large workspace, but their structural flexibility induces vibrations, accelerates fatigue, degrades tracking performance, and limits operational speed. These challenges are further amplified in multi-link serial manipulators, where increased overall length leads to greater structural flexibility. This article presents a backstepping output-feedback framework for fast vibration suppression and tip tracking of an n-degree-of-freedom serial flexible manipulator robot (nDSFMR), with a DeepONet-based approximation for practical deployment. Each link-joint is modeled as a Timoshenko beam coupled with an ODE and transformed into a canonical hyperbolic PDE with boundary dynamics. A backstepping-based boundary controller at the joint is developed to equivalently inject distributed damping along the beam, enabling rapid vibration suppression and trajectory tracking, only using available boundary measurements. To enable real-time implementation and scalability, a DeepONet neural operator is introduced to approximate the backstepping kernels, significantly reducing computational cost and facilitating fast controller updates under varying operating conditions. Experiments on a two-link flexible manipulator demonstrate faster vibration suppression and convergence of the end-effector to the desired trajectory, compared with a linear quadratic regulator (LQR) with feedforward control.

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

The paper extends backstepping boundary control for multi-link flexible manipulators by approximating kernels with DeepONet for real-time use and shows experimental gains over LQR, but approximation effects on stability need checking.

read the letter

Dear colleague, The main point is that this work takes standard backstepping for Timoshenko beam models of flexible links, adds a boundary controller that injects distributed damping from joint measurements, and swaps in DeepONet to approximate the kernels so the thing runs fast enough for multi-link cases and changing conditions. Experiments on a two-link arm report quicker vibration suppression and better tip tracking than LQR with feedforward. What the paper does well is lay out the modeling step from coupled beam-ODE to hyperbolic PDE and keep the controller strictly boundary-based, which matches what is measurable on hardware. The DeepONet move is a straightforward way to cut the usual kernel computation burden and allow updates under varying payloads or trajectories. This specific integration for nDOF serial manipulators counts as new, even if it builds on earlier PDE control results. The soft spots are around the approximation. Backstepping proofs normally need the exact kernel to reach the target damped system; any operator error can leave residual terms that weaken the damping or affect stability margins, especially outside the tested conditions. The abstract gives no error bounds, no ablation on kernel mismatch, and no modified Lyapunov argument for the approximated case, so the experimental win stands but the robustness claim is thinner than it could be. The modeling assumptions are standard for the field and not obviously wrong. This paper is for control researchers working on flexible robotics who already know PDE methods and want to see them made practical with operator learning. A reader in that niche would get value from the framework and the two-link results. It has enough formal grounding and experimental content to deserve a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper develops a backstepping output-feedback boundary controller for an n-degree-of-freedom serial flexible manipulator robot (nDSFMR) modeled as coupled Timoshenko beams and ODEs, transformed to canonical hyperbolic PDEs. The controller injects distributed damping using only boundary measurements to achieve rapid vibration suppression and tip trajectory tracking. For real-time scalability, the backstepping kernels are approximated via a DeepONet neural operator. Experiments on a two-link flexible manipulator show faster suppression and better tracking than LQR with feedforward control.

Significance. If the closed-loop stability and damping-injection properties are preserved under the DeepONet approximation, the framework would provide a practical route to deploying infinite-dimensional PDE control on multi-link flexible robots without requiring full-state sensing or high computational cost. The experimental comparison to LQR supplies concrete evidence of performance gains on hardware.

major comments (2)

[DeepONet approximation and controller implementation] The central practical claim relies on replacing exact backstepping kernels with a DeepONet approximation, yet no Lyapunov analysis, residual-error bound, or closed-loop stability guarantee is supplied for the approximated operator (see the section introducing the DeepONet-based controller and the stability proof for the exact case). Standard backstepping target-system transformation requires the exact kernel to cancel destabilizing terms; any approximation error can leave undamped or unstable modes, especially under payload variation or trajectory changes.
[Experimental validation] The experimental section reports faster vibration suppression than LQR but supplies no quantitative metrics (e.g., settling time, peak overshoot, RMS error), no ablation on kernel approximation error, and no robustness tests under varying payloads or reference trajectories. Without these, it is difficult to assess whether the observed improvement is attributable to the damping-injection property or to other implementation details.

minor comments (2)

[Modeling] Notation for the transformed hyperbolic PDE system and boundary conditions should be introduced with explicit reference to the original Timoshenko variables to improve readability.
[Controller design] The abstract states that the controller uses 'only available boundary measurements,' but the full derivation should clarify which measurements are assumed noise-free and how sensor dynamics are neglected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's detailed review and constructive feedback on our manuscript. We have carefully considered the major comments and provide point-by-point responses below, along with our plans for revisions to address the concerns raised.

read point-by-point responses

Referee: [DeepONet approximation and controller implementation] The central practical claim relies on replacing exact backstepping kernels with a DeepONet approximation, yet no Lyapunov analysis, residual-error bound, or closed-loop stability guarantee is supplied for the approximated operator (see the section introducing the DeepONet-based controller and the stability proof for the exact case). Standard backstepping target-system transformation requires the exact kernel to cancel destabilizing terms; any approximation error can leave undamped or unstable modes, especially under payload variation or trajectory changes.

Authors: We concur that the stability under the DeepONet approximation requires further analysis, as the current proof applies to the exact kernels. The manuscript introduces the DeepONet to enable real-time computation by approximating the kernels, leveraging the operator's ability to learn the mapping from system parameters to kernels. To strengthen this, we will add a subsection providing an error bound for the DeepONet approximation based on its approximation theory, and show that for small enough approximation error, the closed-loop stability is preserved by continuity arguments in the backstepping framework. We will also include simulation results demonstrating stability under small kernel perturbations. revision: yes
Referee: [Experimental validation] The experimental section reports faster vibration suppression than LQR but supplies no quantitative metrics (e.g., settling time, peak overshoot, RMS error), no ablation on kernel approximation error, and no robustness tests under varying payloads or reference trajectories. Without these, it is difficult to assess whether the observed improvement is attributable to the damping-injection property or to other implementation details.

Authors: Thank you for pointing this out. The experimental results in the manuscript qualitatively show faster suppression and better tracking compared to LQR, but we agree that quantitative metrics would enhance the evaluation. In the revised version, we will include tables with settling times, peak overshoots, and RMS errors for vibration suppression and trajectory tracking. We will also add an analysis of the kernel approximation error by comparing DeepONet outputs to exact kernels where possible, and conduct additional tests with varying payloads to assess robustness. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

The paper derives the backstepping boundary controller directly from the Timoshenko beam PDE model transformed to canonical hyperbolic form, using boundary measurements to inject distributed damping. The DeepONet serves only as a practical approximation to the resulting kernels for real-time computation and does not redefine or force the core stability or damping-injection result. Experiments compare against an independent LQR baseline on hardware, providing external falsifiability. No load-bearing step reduces by construction to a fit, self-citation, or input renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; relies on standard domain assumptions for beam modeling and backstepping theory without explicit free parameters or new entities stated.

axioms (1)

domain assumption Timoshenko beam model coupled with ODE accurately captures link-joint dynamics
Invoked to transform the system into a canonical hyperbolic PDE

pith-pipeline@v0.9.0 · 5750 in / 1074 out tokens · 41749 ms · 2026-05-20T12:40:47.259136+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/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

backstepping-based boundary controller at the joint is developed to equivalently inject distributed damping along the beam

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

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[1] [1]

Benosman, and G

M. Benosman, and G. Le Vey, ”Control of flexible manipulators: A survey,”Robotica, 22(5):533-545, 2004

work page 2004

[2] [2]

C. S. Bodley,A digital computer program for the dynamic interaction simulation of controls and structure (DISCOS), 1978, V ol. 2. National Aero. Space Admin., Scien. and Tech. Info. Office

work page 1978

[3] [3]

G. W. Chen, R. Vazquez, and M. Krstic, ”Rapid stabilization of Timoshenko beam by PDE backstepping,”IEEE Trans. Autom. Control, 69(2):1141-1148, 2023

work page 2023

[4] [4]

Cibicik, and O

A. Cibicik, and O. Egeland, ”Kinematics and dynamics of flexible robotic manipulators using dual screws,”IEEE Trans. Robot., 37(1):206- 224, 2020

work page 2020

[5] [5]

J. M. Coron, and B. d’Andrea-Novel, ”Stabilization of a rotating body beam without damping,”IEEE Trans. Autom. Control, 43(5):608-618, 1998

work page 1998

[6] [6]

De Luca, and B

A. De Luca, and B. Siciliano, ”Closed-form dynamic model of planar multilink lightweight robots,”IEEE Trans. Systems, Man, Cyber .: Sys- tems, 21(4):826-839, 1991

work page 1991

[7] [7]

S. S. Ge, T. H. Lee, and G. Zhu, ”Improving regulation of a single-link flexible manipulator with strain feedback,”IEEE Trans. Robot. Autom., 14(1):179-185, 2002

work page 2002

[8] [8]

J. Gu, J. Wang, Z. Liu, M. Tan, J. Yu, and Z. Wu, ”Deformation control and thrust analysis of a flexible fishtail with muscle-like actuation,”IEEE Trans. Robot., 41:159-179, 2024

work page 2024

[9] [9]

B. Z. Guo, and T. T. Meng, ”Robust output regulation for Timoshenko beam equation with two inputs and two outputs,”Int. J. Robust Nonli., 31(4):1245-1269, 2021

work page 2021

[10] [10]

He, and S

W. He, and S. S. Ge, ”Vibration control of a flexible beam with output constraint,”IEEE Trans. Indus. Elect., 62(8):5023-5030, 2015

work page 2015

[11] [11]

H. Wei, S. Zhang, and S. S. Ge, ”Boundary output-feedback stabilization of a Timoshenko beam using disturbance observer,”IEEE Trans. Indus. Elect., 60(11):5186-5194, 2012

work page 2012

[12] [12]

H. Wei, S. Nie, T. Meng, and Y . Liu, ”Modeling and vibration control for a moving beam with application in a drilling riser,”IEEE Trans. Control Systems Tech, 25(3):1036-1043, 2016. AUTHORet al.: TITLE 13

work page 2016

[13] [13]

T. R. Kane, and A. L. David, ”The use of Kane’s dynamical equations in robotics,”The Int. J. Robotics Research, 2(3):3-21, 1983

work page 1983

[14] [14]

Krstic, B

M. Krstic, B. Z. Guo, A. Balogh, and A. Smyshlyaev, ”Control of a tip- force destabilized shear beam by observer-based boundary feedback,” SIAM J. Control Optim., 47(2):553-574, 2008

work page 2008

[15] [15]

Krstic, and A

M. Krstic, and A. Smyshlyaev,Boundary control of PDEs: A course on backstepping designs, 2008, Society for Industrial and Applied Mathematics

work page 2008

[16] [16]

Krstic, A

M. Krstic, A. A. Siranosian, and A. Smyshlyaev, ”Backstepping bound- ary controllers and observers for the slender Timoshenko beam: Part I-Design,”American Control Conference(ACC), pp:2412-2417, 2006

work page 2006

[17] [17]

Krstic, A

M. Krstic, A. A. Siranosian, A. Smyshlyaev, and M. Bement, ”Back- stepping boundary controllers and observers for the slender Timoshenko beam: Part II - Stability and simulations,”In Proc. 45th IEEE Conf. Decis. Control, pp:3938-3943, 2006

work page 2006

[18] [18]

H. H. Lee, ”A new trajectory control of a flexible-link robot based on a distributed-parameter dynamic model,”Int. J. Control, 77(6):546-553, 2004

work page 2004

[19] [19]

B. Li, X. Li, H. Gao, and F. Y . Wang, ” Advances in flexible robotic ma- nipulator systems—Part I: Overview and dynamics modeling methods,” IEEE Trans. Mecha., 29(2):1100-1110, 2024

work page 2024

[20] [20]

B. Li, X. Li, H. Gao, and F. Y . Wang, ” Advances in flexible robotic manipulator systems—Part II: Planning, control, applications, and per- spectives,”IEEE Trans. Mecha., 29(3):1680-1689, 2024

work page 2024

[21] [21]

Lismonde, V

A. Lismonde, V . Sonneville, and O. Br ¨uls, ”A geometric optimization method for the trajectory planning of flexible manipulators,”Multi. System Dynam., 47(4): 347-362, 2019

work page 2019

[22] [22]

Y . Liu, Y . Fu, W. He, and Q. Hui, ”Modeling and observer-based vibration control of a flexible spacecraft with external disturbances,” IEEE Trans. Indus. Elect., 66(11):8648-8658, 2018

work page 2018

[23] [23]

Y . Liu, X. Yao, and W.Zhao, ”Distributed neural-based fault-tolerant control of multiple flexible manipulators with input saturations,”Auto- matica,156:111202, 2023

work page 2023

[24] [24]

Lochan, B

K. Lochan, B. K. Roy, and B. Subudhi, ”A review on two-link flexible manipulators,”Annual Reviews in Control, 42: 346-367, 2016

work page 2016

[25] [25]

T. Long, E. Li, Y . Hu, L. Yang, J. Fan, Z. Liang, and R. Guo, ”A vibration control method for hybrid-structured flexible manipulator based on sliding mode control and reinforcement learning,”IEEE Trans. Neural Networks Learn. Systems, 32(2):841-852, 2020

work page 2020

[26] [26]

Macchelli, C

A. Macchelli, C. Melchiorri, and S. Stramigioli, ”Port-based modeling and simulation of mechanical systems with rigid and flexible links,” IEEE Trans. Robot., 25(5):1016-1029, 2009

work page 2009

[27] [27]

Mattioni, Y

A. Mattioni, Y . Wu, and Y . Le Gorrec, ”Infinite dimensional model of a double flexible-link manipulator: The Port-Hamiltonian approach,” Applied Mathe. Model., 83:59-75, 2020

work page 2020

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