pith. sign in

arxiv: 2607.02167 · v1 · pith:YH23UT64new · submitted 2026-07-02 · 📡 eess.SY · cs.RO· cs.SY

Influence of Radial Basis Activation Functions on Intelligent Controller for Robotic Manipulators

Kimmo Paldanius , Gabriel Da Silva Lima , Wallace Moreira Bessa This is my paper

Pith reviewed 2026-07-03 07:43 UTC · model grok-4.3

classification 📡 eess.SY cs.ROcs.SY
keywords radial basis functionneural network controlrobotic manipulatorsactivation functionsadaptive controltrajectory trackingLyapunov stabilitydisturbance estimation
0
0 comments X

The pith

Activation function selection in RBF networks shapes adaptation dynamics and tracking performance in robotic control.

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

This paper develops an intelligent controller for robotic manipulators that pairs model-based nonlinear control with an RBF neural network approximator to estimate and compensate for uncertainties, friction, and unmodeled dynamics. A Lyapunov-based adaptation law with projection is used to guarantee bounded closed-loop signals and tracking error convergence. The central investigation tests how different radial basis activation functions affect transient behavior, steady-state accuracy, and control smoothness during trajectory tracking experiments on a physical manipulator. Results show stability is preserved across kernels but adaptation speed and practical performance metrics change with the chosen function. This positions activation function selection as a structural design parameter that directly influences closed-loop behavior.

Core claim

Experimental implementation on a robotic manipulator demonstrates that although the Lyapunov-based adaptation law with projection guarantees boundedness of closed-loop signals and convergence of the tracking error for every tested radial basis kernel, activation function selection significantly affects adaptation dynamics and practical tracking performance.

What carries the argument

RBF neural network approximator for online disturbance estimation, where the activation function determines the kernel shape and thereby influences the adaptation process within the combined nonlinear controller.

If this is right

  • Stability and bounded signals are maintained for all radial basis kernels tested.
  • Transient response and steady-state tracking accuracy vary with the activation function.
  • Control signal smoothness changes depending on the chosen kernel.
  • Activation function selection can be treated as a tunable design parameter to optimize practical performance.

Where Pith is reading between the lines

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

  • Similar performance sensitivity to kernel shape may appear in other adaptive controllers that use RBF networks for approximation.
  • Kernel choice could be incorporated into systematic tuning procedures for neural controllers beyond manual trial.
  • The work implies that stability proofs alone are insufficient to predict real-world behavior without testing multiple activation functions.

Load-bearing premise

The Lyapunov-based adaptation law with projection is assumed to guarantee boundedness of closed-loop signals and convergence of tracking error for every tested radial basis kernel, independent of the specific activation function shape.

What would settle it

An experiment on the manipulator where tracking errors diverge or closed-loop signals become unbounded when using one of the radial basis activation functions would show the stability guarantee does not hold independently of kernel shape.

Figures

Figures reproduced from arXiv: 2607.02167 by Gabriel da Silva Lima, Kimmo Paldanius, Wallace Moreira Bessa.

Figure 1
Figure 1. Figure 1: Quanser QArm (digital twin environment). [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Architecture of the RBF neural network. Relying on the assumption that artificial neural networks can perform universal approximation (Scarselli and Tsoi, 1998) with an arbitrary precision εi , it follows that di = ˆd ∗ i + εi , where ˆd ∗ i is the output related with the optimal weight vector w∗ i . The boundedness and convergence properties of the closed-loop signals in the presence of modeling inaccurac… view at source ↗
Figure 3
Figure 3. Figure 3: Results for the sinusoidal function. 4.4 Step-like Tracking (Square Reference) The square reference, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results for the square function. 4.5 Ramp-like Tracking (Triangular Reference) The triangular reference, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results for the triangular function. 4.6 Quantitative Comparison Tables 1–3 summarize the mean tracking metrics for each reference type. For the square reference, overshoot and settling time are included to capture transient response characteristics [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

This paper presents an intelligent control framework for trajectory tracking of robotic manipulators using radial basis function (RBF) neural networks for online disturbance estimation. The proposed control structure combines model-based nonlinear control with an adaptive neural approximator that compensates for parametric uncertainties, friction, and unmodeled dynamics. A Lyapunov-based adaptation law with projection guarantees boundedness of the closed-loop signals and convergence of the tracking error to a compact region. The primary objective of this work is to investigate how the choice of activation function within the RBF network influences transient behavior, steady-state accuracy, and control smoothness. The controller is implemented on a robotic manipulator. Experimental results demonstrate that although stability is preserved for all kernels, activation function selection significantly affects adaptation dynamics and practical tracking performance. These findings demonstrate that activation function selection acts as a structural design parameter in intelligent control, directly shaping adaptation dynamics and practical closed-loop performance.

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.

Referee Report

1 major / 0 minor

Summary. The paper presents an intelligent control framework for trajectory tracking in robotic manipulators that augments a model-based nonlinear controller with an RBF neural network approximator for online compensation of uncertainties, friction, and unmodeled dynamics. A Lyapunov-based adaptation law incorporating a projection operator is used to guarantee bounded closed-loop signals and convergence of the tracking error to a compact set. The central investigation is experimental: while stability is preserved across several radial-basis kernels, the choice of activation function materially affects adaptation transients, steady-state accuracy, and control effort smoothness on a physical manipulator, leading to the claim that activation-function selection functions as a structural design parameter in intelligent control.

Significance. If the uniform-stability premise holds, the work supplies concrete experimental evidence that RBF kernel choice is not merely a tuning detail but a load-bearing structural decision that shapes practical closed-loop behavior. This is a useful contribution to the adaptive-control literature, where most analyses treat the approximator architecture as fixed once the Lyapunov proof is written.

major comments (1)
  1. [stability analysis] Stability analysis (abstract and methods): The manuscript asserts that a single general Lyapunov analysis plus projection operator guarantees boundedness and tracking-error convergence for every tested RBF kernel. However, the ilde{W} term appearing in \dot{V} depends on the approximation-error bound, which is kernel-dependent through Lipschitz constants, support sizes, and conditioning of the regressor oldsymbol{ heta}(x). No kernel-by-kernel re-derivation or uniform bound independent of kernel parameters is supplied, leaving the premise that stability is preserved for all kernels as an unverified assumption rather than a demonstrated result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive comment on the stability analysis. We address the point below.

read point-by-point responses

  1. Referee: Stability analysis (abstract and methods): The manuscript asserts that a single general Lyapunov analysis plus projection operator guarantees boundedness and tracking-error convergence for every tested RBF kernel. However, the ilde{W} term appearing in 𝑉̇ depends on the approximation-error bound, which is kernel-dependent through Lipschitz constants, support sizes, and conditioning of the regressor 𝐃(x). No kernel-by-kernel re-derivation or uniform bound independent of kernel parameters is supplied, leaving the premise that stability is preserved for all kernels as an unverified assumption rather than a demonstrated result.

    Authors: The Lyapunov analysis is formulated generally and relies only on the standard assumptions that the RBF regressor remains bounded (true for all compactly supported radial kernels) and that a finite approximation error bound exists for the chosen network (guaranteed by the universal approximation property of RBFs). The projection operator confines the weight error ilde{W} to a known compact set, rendering its contribution to 𝑉̇ non-positive independently of the specific kernel. This yields 𝑉̇ ≤ -λ||e||^{2} + δ, where δ absorbs the kernel-specific approximation error but preserves the conclusion of uniform ultimate boundedness for each kernel individually. No uniform bound across kernels is claimed or required; the experimental results simply compare performance while confirming that the general proof structure holds for every tested kernel. The analysis is therefore demonstrated rather than assumed. revision: no

Circularity Check

0 steps flagged

No circularity; experimental comparison of RBF kernels on standard adaptive law

full rationale

The paper applies a conventional Lyapunov-based adaptive control law with projection operator to a robotic manipulator and reports experimental outcomes for multiple radial basis activation functions. The stability guarantee is stated as following from the general analysis (independent of specific kernel shape), while observed differences in transient behavior and tracking accuracy are measured directly from hardware trials. No derivation step reduces a claimed prediction to a fitted parameter by construction, no self-citation supplies the central premise, and no ansatz or uniqueness result is imported from prior author work. The derivation chain remains self-contained against the external experimental benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are detailed beyond the standard Lyapunov stability assumption for adaptive control.

axioms (1)
  • domain assumption Lyapunov-based adaptation law with projection guarantees boundedness of closed-loop signals for all RBF kernels
    Invoked in the abstract to ensure stability independent of activation function choice.

pith-pipeline@v0.9.1-grok · 5691 in / 1108 out tokens · 20961 ms · 2026-07-03T07:43:36.804304+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references

  1. [1]

    1991 , publisher=

    Applied nonlinear control , author=. 1991 , publisher=

    1991

  2. [2]

    Neural networks , volume=

    Universal approximation using feedforward neural networks: A survey of some existing methods, and some new results , author=. Neural networks , volume=. 1998 , publisher=

    1998

  3. [3]

    Adaptive Control Tutorial , author =

  4. [4]

    Journal of the Brazilian Society of Mechanical Sciences and Engineering , volume=

    Accurate trajectory tracking control with adaptive neural networks for omnidirectional mobile robots subject to unmodeled dynamics , author=. Journal of the Brazilian Society of Mechanical Sciences and Engineering , volume=. 2023 , publisher=

    2023

  5. [5]

    IEEE Access , volume=

    Robust finite-time adaptive nonlinear control system for an EOD robotic manipulator: Design, implementation, and experimental validation , author=. IEEE Access , volume=. 2024 , publisher=

    2024

  6. [6]

    IEEE Transactions on Automation Science and Engineering , volume=

    Neural Adaptive Finite-Time Tracking Control for Strict-Feedback Nonlinear Systems and Application to Robotic Manipulators , author=. IEEE Transactions on Automation Science and Engineering , volume=. 2025 , publisher=

    2025

  7. [7]

    Journal of Automation and Intelligence , volume=

    Decentralised adaptive learning-based control of robot manipulators with unknown parameters , author=. Journal of Automation and Intelligence , volume=. 2025 , publisher=

    2025

  8. [8]

    2026 , publisher=

    Model with Neural Network Component for Adaptive Manipulator Control under Variable Load , author=. 2026 , publisher=

    2026

  9. [9]

    IEEE Transactions on Artificial Intelligence , volume=

    Adaptive intelligent control-based consensus tracking for a class of switched nonstrict feedback nonlinear multiagent systems with unmodeled dynamics , author=. IEEE Transactions on Artificial Intelligence , volume=. 2023 , publisher=

    2023

  10. [10]

    International Journal of Automotive Technology , pages=

    An RBFNN-Optimized Active Disturbance Rejection Controller for Path Tracking of Autonomous Vehicles , author=. International Journal of Automotive Technology , pages=. 2026 , publisher=

    2026

  11. [11]

    The Open Civil Engineering Journal , volume=

    A review on adaptive methods for structural control , author=. The Open Civil Engineering Journal , volume=

  12. [12]

    Annual Reviews in Control , volume=

    A review on model reference adaptive control of robotic manipulators , author=. Annual Reviews in Control , volume=. 2017 , publisher=

    2017

  13. [13]

    Annual Review of Control, Robotics, and Autonomous Systems , volume=

    Safe learning in robotics: From learning-based control to safe reinforcement learning , author=. Annual Review of Control, Robotics, and Autonomous Systems , volume=. 2022 , publisher=

    2022

  14. [14]

    IEEE/ASME Transactions on Mechatronics , year=

    Learning-Based Methods for Aerial Manipulation: A Focused Review , author=. IEEE/ASME Transactions on Mechatronics , year=