Learning-Based Progressive Barrier Control for Robot Manipulators with Initial Errors Outside Prescribed Tracking Bounds

Hamed Rahimi Nohooji; Danial Zafaranchizadeh Moghaddam; Abolfazl Zaraki; Holger Voos

cs.ROMar 18, 2025

Learning-Based Progressive Barrier Control for Robot Manipulators with Initial Errors Outside Prescribed Tracking Bounds

Authors: Hamed Rahimi Nohooji, Danial Zafaranchizadeh Moghaddam, Abolfazl Zaraki, Holger Voos

Organizations: Automation and Robotics Research Group, Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, 1855 Luxembourg, Luxembourg · School of Physics, Engineering and Computer Science (SPECS), and Robotics Research Group of the University of Hertfordshire, Hatfield, AL10 9AB, United Kingdom

Abstract

Robot manipulators may start a new task with a tracking error larger than the prescribed tolerance. Conventional barrier controllers generally require the initial error to lie within this tolerance, which prevents their direct use under such conditions. This paper develops a progressive barrier controller that gradually contracts an initial error bound to the required value within a prescribed time. The robot can therefore start outside the final bound, while the direct joint-position error satisfies it after the transition. The closed-form control law combines progressive barrier feedback with an online adaptive torque term based on an actor--critic structure. A Lyapunov analysis establishes bounded closed-loop signals and gives sufficient conditions for satisfaction of the final tracking bound. Two-link simulations consider large initial errors, actuator saturation, dynamic variations, disturbances, and measurement errors. The adaptive term reduces the median root-mean-square tracking error by 45.1% compared with the zero-weight progressive barrier controller. The simulations also map the initial errors that can be handled at different transition times under fixed torque limits. Finally, an experiment on a Niryo Ned3 Pro illustrates tracking performance using measured position and motor-current data.

Figures & tables

Fig. 1: Progressive imposition of the joint tracking bound. The photograph at left shows the Niryo Ned3 Pro used in the single-joint experiment. The plot at right shows the nominal two-link simulation. The direct tracking error may initially lie outside the final bound, whereas the scaled error starts inside the barrier domain. The two errors coincide after the prescribed transition time.

Case	Link-2 mass increment (kg)	Friction scale	Disturbance scale	Measurement amplitude (rad)	Phase offset (rad)	Reference-speed scale
1	0.00	0.80	0.50	0.0000	0.00	0.80
2	0.25	0.95	1.25	0.0008	0.35	1.06
3	0.50	1.10	0.97	0.0016	0.70	0.89
4	0.00	1.24	0.68	0.0004	1.05	1.15
5	0.25	0.86	1.44	0.0012	1.40	0.99
6	0.50	1.01	1.15	0.0020	1.75	0.82

TABLE I: Fixed Paired Perturbation Cases

Fig. 2: Feasibility under the same torque limits. Each marker represents one proposed-controller run, and failed settings are retained. Longer transition times accommodate larger initial errors in this grid.

Fig. 3: Nominal two-link response from an initial tracking error outside the final bound: (a) joint tracking, (b) direct and scaled normalized tracking errors, and (c) applied joint torques under the fixed limits. The vertical dotted line marks $T_{c}$ , and the shaded interval denotes the prescribed transition. The horizontal line in panel (b) marks $\rho=1$ , while those in panel (c) mark the two joint torque limits.

Fig. 4: Paired results for the 12 fixed perturbation cases: (a) vector-error RMS for

t\geq T_{c}

, (b) largest normalized error for

t\geq T_{c}

, and (c) last entry into the final envelope. Markers denote individual cases; boxes span the 25th–75th percentiles, center lines denote medians, and whiskers show the full range. CT denotes computed torque; lower values are better.

Fig. 5: Complete single-joint physical record without smoothing: reference and measured joint angle, error recomputed with the paper’s sign convention

q-q_{d}

, and commanded and measured current in logged raw units.

Controller	Successful cases	Median $e_{\rm rms}$ (rad)	Worst $\rho$ , $t\geq T_{c}$	Median last entry (s)
Proposed	12/12	0.0250	0.075	0.258
Progressive barrier without learning	12/12	0.0456	0.131	0.258
Computed torque	12/12	0.0498	0.243	0.421

TABLE II: Two-Link Results Under Torque Limits

Interval	MAE	RMS	95th pct.	Max.
Complete record	0.957	1.154	2.141	3.520
20–120 s	0.940	1.125	2.089	2.948

TABLE III: Single-Joint Tracking-Error Statistics (deg)

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This paper addresses the challenge of simultaneously compensating for state-dependent uncertainties and enforcing time-varying state constraints in Euler-Lagrange systems, a common requirement in robotics that remains underserved by existing control designs. A novel adaptive control framework is developed that combines an artificial time-delay-based uncertainty estimation strategy, also known as time-delay estimation, with a barrier Lyapunov function to enforce constraint-aware control design. Specifically, a state-dependent upper bound on the time-delay estimation approximation error is analytically formulated, and an adaptive law is constructed to estimate its parameters online, enabling real-time state-dependent uncertainty compensation without relying on prior model knowledge. To ensure constraint compliance, the barrier Lyapunov function-based controller enforces time-varying bounds on both position and velocity. The resulting architecture is provably stable via Lyapunov analysis. Experimental results on a five-degree-of-freedom robotic manipulator validate the framework's capability, compared with the state of the art, in maintaining strict adherence to safety-critical constraints under dynamic uncertainties.

Abstract

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