Adaptive Compensation of Nonlinear Friction in Mechanical Systems Without Velocity Measurement

Alexey Bobtsov; Jose Guadalupe Romero; Leyan Fang; Romeo Ortega

REVIEW 4 major objections 5 minor 1 cited by

Adaptive Compensation of Nonlinear Friction in Mechanical Systems Without Velocity Measurement

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that friction in a mechanical system can be compensated adaptively using only position measurements, with the speed estimate converging for all initial conditions.

desk verdict A solid I&I observer for stiction-plus-Coulomb friction without velocity measurement, with a local proof error and a tracking claim that is conditional on an unverified excitation assumption. read the letter →

arxiv 2508.00175 v1 pith:5X5KO5O6 submitted 2025-07-31 eess.SY cs.SY

Jose Guadalupe Romero , Romeo Ortega , Leyan Fang , Alexey Bobtsov This is my paper · ORCID

classification eess.SYcs.SY MSC 93C4093B0793C10

keywords adaptivefrictioncompensationimmersionandinvariancevelocityobserverCoulombstictionglobalconvergencetrackingcontrolLuGremodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper takes on a long-standing control problem: canceling friction in a mechanical system without any velocity measurement and without knowing the friction parameters. It proposes an immersion-and-invariance adaptive observer that estimates both the speed and the two unknown friction coefficients from position measurements alone, and proves the speed estimate converges to the true velocity for all initial conditions whenever the state remains bounded. On top of this observer, a certainty-equivalent position controller is shown to achieve global tracking of a desired reference, provided a state-dependent excitation condition on the estimated regressor holds. If the claims stand, servo drives and robotic joints could drop velocity sensors and still get precise motion control under stiction-plus-Coulomb friction.

What carries the argument

The engine is an immersion-and-invariance construction that splits each estimate into a proportional plus an integral part: x̂2=x2I+k1x1, θ̂1=θ1I-(ϑ/(2k1))x̂$2^{2}$, θ̂2=θ2I-(1/k1)log($\cosh$(ϑx̂2)). These choices make the estimation-error dynamics admit the Lyapunov function H=½(ϑx̃$2^{2}$+θ̃$1^{2}$+θ̃$2^{2}$), whose time derivative is dominated by a strictly negative term in x̃2 because the tanh function is strongly monotone, and a standard convergence lemma yields x̃2→0. For the tracking result, the same error system is rearranged into a linear time-varying form whose unforced part is globally asymptotically stable under the excitation condition adapted from the cited stability theorem.

What would settle it

Run the closed-loop system with a constant reference r, which makes the estimated regressor φ(t)=[x̂2(t), tanh(ϑx̂2(t))]^T non-exciting under the paper's own discussion. If parameter estimates stay bounded but do not converge and the tracking error does not vanish, the excitation assumption is load-bearing for the global tracking claim; conversely, if tracking still succeeds, the need for Assumption 1 is weakened and the global claim may hold in greater generality.

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Extended reading notes

Core claim

The central claim is that, for the plant ẋ1=x2, ẋ2=-θ1x2-θ2 tanh(ϑx2)+u where only x1 is measured and ϑ is known, the proposed I&I observer guarantees lim_{t→∞}(x̂2(t)-x2(t))=0 for all initial conditions, with all signals bounded, making it the first globally convergent adaptive friction compensator without velocity measurement for this model. The tracking extension states that with the same observer feeding a certainty-equivalent control law, and under Assumption 1 (non-summable excitation of the estimated regressor [x̂2, tanh(ϑx̂2)]^T), both parameter estimates and the tracking error converge to zero over time.

Load-bearing premise

The global tracking claim rests entirely on Assumption 1, a state-dependent excitation condition on the estimated speed signal that the paper itself calls 'rather cryptic' and never verifies for any concrete reference; if the condition fails, parameter estimates are not proven to converge and the tracking guarantee disappears.

Editorial extensions

If this is right

A servo drive or robot joint could run the proposed controller with only a position encoder, avoiding the cost, wiring, and noise of tachometers.
The observer result holds for all initial conditions, not just local or small-error configurations, when the state remains bounded.
The tracking result requires a weaker excitation condition than classic persistent excitation, so references with interspersed quiet intervals may still satisfy it.
The same observer shape extends to a three-state hydro-mechanical system with an additional pressure state, requiring only an upper bound on one friction parameter.

Reading between the lines

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

Because the proof only uses strong monotonicity of tanh, the observer should generalize to any friction nonlinearity that is strongly monotone in velocity, such as other smooth approximations of the sign function.
The state-dependent excitation condition could be turned into a practical design tool: before commissioning, simulate the reference trajectory and check numerically whether the interval sums in (20)-(21) grow, which would give engineers a verifiable certificate for the tracking claim.
The concluding remarks leave dynamic friction models (LuGre, Dahl, Stribeck) as an open problem; this paper's LuGre simulations suggest robustness to model mismatch, but no proof, so a natural next test is the same observer against the full LuGre model with uncertainty in its internal state dynamics.
The 'global' qualifier should be read as global in initial conditions under an input that keeps the state bounded; the observer proof itself assumes boundedness of the state, so the design does not yet handle inputs that drive the system to infinity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new piece is the adaptive speed observer of Proposition 1, which extends the authors' I&I line to include an unknown Coulomb coefficient θ2. That is a real step beyond their earlier stiction-only observers, and the construction is elegant: the tanh term is absorbed by θ2P(x̂2) = −(1/k1) log(cosh(ϑx̂2)). The Lyapunov argument nearly works, but there is a concrete mistake: tanh is not strongly monotone on all of R in the sense of Definition 1, so the bound in (14) is false as written. The weaker bound Ḣ ≤ −γ1ϑx̃2² still gives x̃2 → 0 via Barbalat's lemma, so the observer claim stands with a minor repair.

The tracking result is another matter. Proposition 2 is lifted from the co-author's [26] and depends on Assumption 1, a state-dependent condition on the regressor φ(t) that the paper itself calls 'rather cryptic.' The stress-test observation is on point: for the step-plus-ramp simulation, once the ramp is reached φ(t) tends to a constant vector, so the integral in (20) becomes asymptotically rank-one and the λk² sum may fail to diverge. That places the paper's own demonstration outside the theorem's hypotheses. Also, the passage from the unforced system's global asymptotic stability to the forced system with the σ(t) term is asserted without a uniformity argument. These are fixable but matter: the headline claim of a globally convergent tracking controller is conditional, and the actual scope is narrower than the abstract suggests.

Credit where due: the paper is clearly written, honest about the cryptic excitation condition, and the simulations—especially the LuGre robustness test—are useful. The citation pattern is fine; [26] is a co-authored reference, but the import is explicitly flagged and the observer result does not depend on it.

This deserves a serious referee. The observer result is worth publishing after the monotonicity slip is corrected; the tracking proposition needs either a verifiable sufficient condition or a clearer statement of what remains open. I would suggest a major revision, not a desk reject.

Referee Report

4 major / 5 minor

Summary. This manuscript treats a single-degree-of-freedom mechanical system with linear viscous damping plus a smooth Coulomb/stiction friction term, with only the position measured and the friction parameters unknown. The authors propose an immersion-and-invariance adaptive observer that estimates the velocity and the two friction parameters, and they prove convergence of the velocity estimate. A certainty-equivalent tracking controller is then added, and global tracking is claimed under an excitation assumption (Assumption 1) imposed on the closed-loop regressor. The paper also adapts the observer to a third-order hydro-mechanical model and presents simulations with a LuGre friction model, including a step-plus-ramp reference signal. The abstract advertises this as the first globally convergent solution to adaptive friction compensation without velocity measurement.

Significance. If the tracking result is made fully rigorous, this would be a valuable contribution: the observer is a transparent I&I construction with direct Lyapunov cancellations and no fitted parameters, and the simulations suggest robustness to the LuGre model mismatch. However, the global tracking theorem is conditional on a state-dependent excitation condition whose verification is not supplied, and its proof is delegated to a conference paper rather than derived in the manuscript. The false strong-monotonicity inequality in the observer proof is localized and repairable, but the tracking claim needs additional work before the advertised result is supported.

major comments (4)

[Section II-B, Definition 1 and Eq. (14)] The mapping tanh does not satisfy the strong-monotonicity Definition 1 on all of R, since tanh'(s)=sech^2(s) tends to 0 as |s| tends to infinity. Consequently the inequality −θ2(ϑx̂2−ϑx2)[tanh(ϑx̂2)−tanh(ϑx2)] ≤ −θ2ϑ^2 x̃2^2, used to obtain (14), is not valid as written. The weaker inequality −θ2(ϑx̂2−ϑx2)[tanh(ϑx̂2)−tanh(ϑx2)] ≤ 0 is available, which still yields ᵊċ ≤ −γ1ϑx̃2^2. That weaker bound is enough, together with boundedness of ᵋdx̃2 from (6), to conclude x̃2∈L^2∩L∞ and hence x̃2(t)→0 via Barbalat's lemma. The proof of Proposition 1 should be corrected to state this weaker bound and to adjust the final argument accordingly.
[Section III-B, Assumption 1 and Proposition 2] Assumption 1 is a condition on the closed-loop regressor φ(t)=[x̂2(t), tanh(ϑx̂2(t))]^T, not a condition on the reference r(t) alone, and it is never verified for any of the simulations. In the step-plus-ramp example of Section V, once the ramp is reached x̂2 tends to a constant v*, so φ(t) tends to the constant vector [v*, tanh(ϑv*)]^T; the integral in (20) then becomes asymptotically rank-one and the λk will decay at the rate of the error convergence. Whether (21) holds is not demonstrated, so the paper's own step-plus-ramp demonstration may lie outside the hypotheses of Proposition 2. The authors should either verify Assumption 1 for the examples or explicitly state that those simulations are heuristic illustrations rather than instances of the theorem.
[Section III-B, proof of Proposition 2] The proof of Proposition 2 is not self-contained and contains a nontrivial gap. It is asserted that because σ(t) is bounded and converges to zero, and the unforced part of (22) is globally asymptotically stable, it is sufficient to ensure χ(t)→0. For time-varying systems this is not automatic: a vanishing perturbation can destroy global asymptotic stability unless a uniformity or robustness property is established. The manuscript should either prove this perturbation step directly or state the precise theorem from [26] that justifies it and verify that its hypotheses (including any boundedness and regularity conditions on φ) hold in the present closed loop. In addition, Proposition 1 assumes u is such that the state remains bounded, but Proposition 2 does not prove this boundedness for the closed loop (17); a bootstrap argument is needed before the observer result can be invoked.
[Section IV, Proposition 3 and Eq. (31)-(33)] The Lyapunov function in (31) appears to be missing a square: it should be U = H + (1/2)α1 x̃3^2 rather than U = H + (1/2)α1 x̃3. More importantly, the derivative in (33) contains the term −α1 a3 x̃3^2, but the displayed inequality later contains −α1 a1 x̃3^2 and the constant α3 is defined as α1 a1−1. Unless a3=a1 in the system (26), condition (32) with a1 does not ensure α3>0 as written. This affects the stated result of Proposition 3 and needs to be corrected.

minor comments (5)

[Section II-B heading] The heading contains a typo: “Adaptive obsesrver” should be “Adaptive observer”.
[Section V, Fig. 5] The caption says “PE condition under a cosine reference signal” but the vertical axis is not labeled and no scalar measure of excitation is defined; please clarify what quantity is plotted.
[Section III-B and Section V] The proof of Proposition 2 is delegated to [26] without stating the assumptions or the theorem; for a journal paper the key argument should be reproduced or at least precisely stated, especially since [26] is not a standard textbook reference.
[Section VI] The concluding remarks state that this is the “first solution” of the problem, but the global tracking claim is conditional on Assumption 1; the conclusion should be qualified accordingly.
[Eq. (35)] The notation in (35) reuses the symbols θ̃1 and θ̃2 for errors relative to the LuGre parameters σ2 and FC, after these symbols were defined for the stiction-plus-Coulomb model; this overloading should be made explicit to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the observer is built by direct Lyapunov cancellation, and the tracking result is conditional on an imported weak-excitation theorem rather than on the paper's own conclusion.

full rationale

The core observer construction in Proposition 1 is self-contained: the I&I mappings are chosen so that the cross-terms in the Lyapunov derivative cancel, yielding ˙H ≤ −γ1 ϑ x̃2^2 (the claimed stronger bound (14) is not needed for this). No fitted parameter is renamed as a prediction; the tuning gain k1 is a design parameter, and the parameter estimates are genuine unknowns. Proposition 2 is conditional on Assumption 1, an excitation condition on the closed-loop regressor φ(t). The proof is delegated to Proposition 1 of [26], a co-authored external theorem, but this is a citation of an independent mathematical result, not a reduction of the claim to its own inputs: Assumption 1 does not contain the target tracking conclusion, and the theorem is parameter-free with stated hypotheses that do not include the friction-tracking result. The paper would be stronger if it verified Assumption 1 for its examples and supplied the missing perturbation argument for the σ(t) term, but those are completeness and correctness concerns, not circularity. The false uniform strong-monotonicity bound in (14) is a localized mathematical error that does not affect the needed non-positive sign of the tanh cross term, and it is likewise not a circularity issue.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants; the unknown friction coefficients are estimated online. The main load-bearing premises are the boundedness of the closed-loop state, the false strong-monotonicity of tanh, the known slope ϑ, and the unverifiable excitation Assumption 1 for tracking.

assumptions (5)

ad hoc to paper tanh satisfies Definition 1 with a uniform c>0
The proof of Proposition 1 uses this in inequality (14); it is false because tanh'(x)=sech^2(x) has infimum 0. The conclusion might still follow from monotonicity, but the printed argument is invalid.
domain assumption The state remains bounded under the control (bounded u)
Propositions 1 and 3 assume u such that the state remains bounded; for the closed-loop tracking controller, boundedness is not separately proven before applying the observer.
domain assumption Reference r satisfies Assumption 1 (weak excitation)
Proposition 2 is conditional on inequalities (20)-(21) on φ(t); the condition is state-dependent, called 'rather cryptic' in Section III-C, and no constructive check is provided.
domain assumption The tanh slope ϑ is known and sufficiently large
Footnote 4 treats ϑ as a known large smoothing parameter; the proof does not quantify how large it must be for the tanh approximation to the relay friction.
domain assumption Motor inertia is known and lumped into u, θ1, θ2
Section II-A states this without loss of generality; it restricts the system class covered by the theorem.

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Cite this review

Pith. "Pith review of Adaptive Compensation of Nonlinear Friction in Mechanical Systems Without Velocity Measurement." pith.science (2026). https://pith.science/paper/5X5KO5O6

@misc{pith2026250800175,
  author       = {Pith},
  title        = {Pith review of: Adaptive Compensation of Nonlinear Friction in Mechanical Systems Without Velocity Measurement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5X5KO5O6}},
  note         = {Machine review of arXiv:2508.00175}
}

read the original abstract

Friction is an unavoidable phenomenon that exists in all mechanical systems incorporating parts with relative motion. It is well-known that friction is a serious impediment for precise servo control, hence the interest to devise a procedure to compensate for it -- a subject that has been studied by many researchers for many years. The vast majority of friction compensation schemes reported in the literature rely on the availability of velocity measurements, an information that is hard to obtain. A second limitation of the existing procedures is that they rely on mathematical models of friction that contain several unknown parameters, some of them entering nonlinearly in the dynamic equations. In this paper we propose a globally convergent tracking controller for a mechanical system perturbed by static and Coulomb friction, which is a reliable mathematical model of the friction phenomenon, that does not rely one measurement of velocity. The key component is an immersion and invariance-based adaptive speed observer, used for the friction compensation. To the best of our knowledge, this is the first globally convergent solution to this challenging problem. We also present simulation results of the application of our observer for systems affected by friction, which is described by the more advanced LuGre model.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers
eess.SY 2026-08 conditional novelty 6.0 of 10

The 4.3x ISE gap between the compared observers is the tracking-error response of the chosen poles (zeta=13.7), not an observer property; properly tuned super-twisting wins on hardware.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [26]

Barabanov and R

N. Barabanov and R. Ortega, On global asymptotic stabil ity of SPR adaptive systems without persistent excitation, 56th IEEE Conference on Decision and Control , Melbourne, Australia, December 12-15, 2017

work page 2017

[2] [1]

Dahl, Solid friction damping of mechanical vibration s, AIAA Journal, vol

P . Dahl, Solid friction damping of mechanical vibration s, AIAA Journal, vol. 40, no. 12, pp. 1675-1682, 1976

work page 1976

[3] [2]

Canudas, H

C. Canudas, H. Olsson, K. J. Astrom and P . Lischinsky, A ne w model for control of systems with friction, IEEE Trans. Automat. Control , vol. 40, 1995

work page 1995

[4] [3]

Armstrong, P

B. Armstrong, P . Dupont and C. Canudas, A survey of analys is tools and compensation methods for the control of machines with fr iction, Automatica, vol. 30, pp. 1083-1138, 1994

work page 1994

[5] [4]

Ruderman, Analysis and Compensation of Kinetic Friction in Robotic and Mechatronic Control Systems , CRC Press, 2024

M. Ruderman, Analysis and Compensation of Kinetic Friction in Robotic and Mechatronic Control Systems , CRC Press, 2024

work page 2024

[6] [5]

Winston and P

G. Winston and P . Gilbert, On adaptive friction compensa tion, Automat- ica, 1974

work page 1974

[7] [6]

I. D. Landau, Adaptive Control: The Model Reference Approach , Marcel Dekker, NY , 1979

work page 1979

[8] [7]

Canudas and P

C. Canudas and P . Lischinsky, Adaptive friction compens ation with partially known dynamic friction model, Int. J. Adaptive Control and Sign. Proc. , vol. 11, pp. 65–80, 1991

work page 1991