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

Carlos MUÑIZ-MONTERO; Luis A. SÁNCHEZ-GASPARIANO; Javier LEMUS-LÓPEZ

doi:10.2478/ama-2025-0070

Figures & Tables

(a) Two-DOF manipulator diagram; (b) Desired positions in the workspace for RBF training

Interpolation of kp1(qd1, qd2) and kp2(qd1, qd2)

Responses for qd1 = –40° and qd2 = 120°. (a) Position errors. (b) Comparison of ℒ2 Norms

Responses for qd1 = –40° and qd2 = 120°. (a) Torque responses. (b) Angular velocities

Monte Carlo robustness of PD, Tanh, and PDN controllers under parametric uncertainty and (qd1, qd2) = (–40°, 120°)

Boxplots of robustness metrics (Ts, e(ts), ISE, and ℒ2 – i. e., ∥e∥2 –) for PD, Tanh, and PDN controllers with (qd1, qd2) = (–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.

Parameters of the simulated anthropomorphic arm manipulator

Parameter	Value
l₁, l₂ (link lengths)	0.45m
τ1max (shoulder), τ2max (elbow)\tau _1^{max}{\rm{ (shoulder), }}\tau _2^{max}{\rm{ (elbow)}}	150 Nm, 15 Nm
k_g₁(q), k_g₂(q)	40.28 Nm, 1.81 Nm
Inertia Matrix	M(q)=[ 2.351+0.167cos(q2)0.102+0.083cos(q2)0.102+0.083cos(q2)0.102 ]\matrix{ {M(q)} \hfill & = \hfill & {\left[ {\matrix{ {2.351 + 0.167\cos \left( {{q_2}} \right)} \hfill & {0.102 + 0.083\cos \left( {{q_2}} \right)} \hfill \cr {0.102 + 0.083\cos \left( {{q_2}} \right)} \hfill & {0.102} \hfill \cr } } \right]} \hfill \cr }
Coriolis Matrix	C(q,q˙)=[ −0.1676sin(q2)q˙2−0.083sin(q2)q˙20.084sin(q2)q˙10 ]\matrix{ {C(q,\dot q)} \hfill & = \hfill & {\left[ {\matrix{ { - 0.1676\sin \left( {{q_2}} \right){{\dot q}_2}} \hfill & { - 0.083\sin \left( {{q_2}} \right){{\dot q}_2}} \hfill \cr {0.084\sin \left( {{q_2}} \right){{\dot q}_1}} \hfill & 0 \hfill \cr } } \right]} \hfill \cr }
Gravitational torque vector	g(q)=9.81[ 3.92sin(q1)+0.186sin(q1+q2)0.186sin(q1+q2) ]g(q) = 9.81\left[ {\matrix{ {3.92\sin \left( {{q_1}} \right) + 0.186\sin \left( {{q_1} + {q_2}} \right)} \cr {0.186\sin \left( {{q_1} + {q_2}} \right)} \cr } } \right]
Friction coefficient matrix	B=[ 2.288000.175 ]B = \left[ {\matrix{ {2.288} & 0 \cr 0 & {0.175} \cr } } \right]

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

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

Figures & Tables

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Parameters of the simulated anthropomorphic arm manipulator

Comparative table of studies on PD controllers with variable gains

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