PD Control with Auto-Tuned Gains Using Rbf Networks for Enhanced Trajectory Tracking in Manipulator Robots

Parameter	Value
l₁, l₂ (link lengths)	0.45m
$τ_{1}^{m a x} (shoulder), τ_{2}^{m a x} (elbow)$	150 Nm, 15 Nm
k_g₁(q), k_g₂(q)	40.28 Nm, 1.81 Nm
Inertia Matrix	$\begin{array}{l} M (q) & = & [\begin{array}{l} 2.351 + 0.167 \cos (q_{2}) & 0.102 + 0.083 \cos (q_{2}) \\ 0.102 + 0.083 \cos (q_{2}) & 0.102 \end{array}] \end{array}$
Coriolis Matrix	$\begin{array}{l} C (q, \dot{q}) & = & [\begin{array}{l} - 0.1676 \sin (q_{2}) {\dot{q}}_{2} & - 0.083 \sin (q_{2}) {\dot{q}}_{2} \\ 0.084 \sin (q_{2}) {\dot{q}}_{1} & 0 \end{array}] \end{array}$
Gravitational torque vector	$g (q) = 9.81 [\begin{matrix} 3.92 \sin (q_{1}) + 0.186 \sin (q_{1} + q_{2}) \\ 0.186 \sin (q_{1} + q_{2}) \end{matrix}]$
Friction coefficient matrix	$B = [\begin{matrix} 2.288 & 0 \\ 0 & 0.175 \end{matrix}]$

Interpolation of k_p1(q_d1, q_d2) and k_p2(q_d1, q_d2)

Responses for q_d1 = –40° and q_d2 = 120°. (a) Position errors. (b) Comparison of ℒ₂ Norms

Responses for q_d1 = –40° and q_d2 = 120°. (a) Torque responses. (b) Angular velocities

Monte Carlo robustness of PD, Tanh, and PDN controllers under parametric uncertainty and (q_d1, q_d2) = (–40°, 120°)

Boxplots of robustness metrics (*T_s*, e(*t_s*), ISE, and ℒ₂ – i. e., ∥e∥² –) for PD, Tanh, and PDN controllers with (q_d1, q_d2) = (–40°, 120°), ±15% parametric uncertainty, and 200 Monte Carlo trials

Disturbance Rejection to a 6 Nm, 0.15 s Torque Pulse at Joint 2 (t = 3 s): q1 and q2 Position Errors for PD, Tanh, and PDN

Point-to-point "Owl" trajectory tracking. (a) Ideal trajectory. (b) Trajectory with the PDN controller. (c) Actuator speed with the Tanh controller. (d) Actuator speed with the PDN controller. (e) Comparison of ℒ₂ norms. (f) Comparison of energy consumption.

Tab. 2.

Comparative table of studies on PD controllers with variable gains

Study	Controller Type	Variable Gain Structure	Stability Analysis Method	Validation Approach
[4]	PD-like with variable gains	Variable; state-, position-, and velocity-dependent; smooth functions (e.g. cos² (tanh (error+velocity)))	Lyapunov theory; global asymptotic stability; gravity compensation required	Simulation; two-DOF direct-drive robot; joint regulation; L2 norm
[15]	PD iterative neural-network learning (PDISN)	Likely variable/adaptive; neural network and iterative learning	Extended Lyapunov theories; stability type not specified	Simulation; manipulator characteristics not specified; scenario not specified
[16]	Proportional-derivative (PD)	Variable; tuned by self-organizing fuzzy algorithm	Not analyzed (no details)	Simulation; manipulator characteristics not specified; tracking control; position error metric
[17]	Self-tuning PD	Bounded, time-varying; neurofuzzy recurrent scheme	Lyapunov theory; semi-global exponential stability	Simulation; manipulator characteristics not specified; trajectory tracking
[18]	Nonlinear PID with fuzzy self-tuned PD gains	Variable, position-dependent; fuzzy logic	Not mentioned; global asymptotic stability; no gravity compensation	Experiments; type not specified; scenario and metrics not specified
[19]	Adaptive PD	Adaptive to gravity parameters	Not mentioned; global convergence	Simulation; three-DOF manipulator; point-to-point and tracking
[20]	PD-type robust	Variable, error-varying; parameterized by perturbing parameter	Singular perturbation theory; stability type not explicit	Physical experiment; planar two-DOF direct-drive robot; trajectory tracking
[21]	Adaptive iterative learning control (ILC)-PD	Variable; iterative learning, two iterative variables	Lyapunov theory; asymptotic convergence	Simulation; two-DOF manipulator; trajectory tracking
[22]	PD-type	Variable, state-dependent	Not mentioned; global asymptotic stability claimed	Physical experiment; two-DOF directdrive arm; scenario not specified
[23]	Linear and nonlinear PD-type	Nonlinear functions of system states	Not mentioned; global asymptotic stability claimed	Simulation; single-link and two-DOF robots; trajectory tracking
This work	PD-like with variable gains	Variable; desired position dependent proportional gains with RBF interpolation networks trained offline	Lyapunov theory; global asymptotic stability; gravity compensation required	Simulation; two-DOF direct-drive robot; joint regulation; L2 norm, point-to-point tracking; regulation performance evaluated with parametric uncertainties and external perturbations

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DOI: https://doi.org/10.2478/ama-2025-0070 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 617 - 625

Submitted on: Sep 21, 2024

Accepted on: Oct 5, 2025

Published on: Dec 19, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

PD control,

Radial Basis Function Networks,

trajectory tracking,

Lyapunov stability

Related subjects:

Engineering,

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Carlos MUÑIZ-MONTERO, Luis A. SÁNCHEZ-GASPARIANO, Javier LEMUS-LÓPEZ, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 4 (December 2025)

PD Control with Auto-Tuned Gains Using Rbf Networks for Enhanced Trajectory Tracking in Manipulator Robots

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