Adaptive Artificial Time-Delay Control with Barrier Lyapunov Constraints for Euler-Lagrange Robots

The evolution of robotic control has transitioned from classical linear controllers to sophisticated nonlinear and adaptive strategies. Early methods like PID control and linear feedback control proved effective in simple, well-modeled environments but faced limitations with complex, uncertain dynamics. The advent of Lyapunov-based adaptive control, exemplified by Slotine and Li’s work, allowed for stability guarantees even with parametric uncertainties. Subsequently, robust control techniques such as H∞ and sliding mode control addressed disturbances but often at the cost of conservativeness or chattering issues. Model predictive control (MPC) emerged as a promising approach for handling constraints explicitly, yet its computational demands hinder real-time deployment in embedded systems. Recently, barrier Lyapunov functions (BLF) have gained attention for their ability to enforce state constraints directly within the control design, providing safety guarantees without online optimization. Meanwhile, artificial time delay estimation (TDE) offers a low-cost, model-free means to approximate unknown dynamics by leveraging delayed measurements. Despite these advances, integrating TDE with BLF to simultaneously address dynamic uncertainties and safety constraints remains an open challenge, especially for nonlinear Euler-Lagrange systems with state-dependent uncertainties.

In practical robotic applications, uncertainties in system parameters and external disturbances pose significant challenges to achieving precise and safe motion control. Traditional adaptive controllers often rely on prior knowledge of bounds or assume parametric uncertainties are fixed, which is unrealistic in dynamic environments. Moreover, robots operating near humans, in constrained spaces, or handling delicate payloads require strict enforcement of position and velocity limits to prevent accidents or damage. Existing methods either focus on uncertainty compensation without explicit constraint handling or enforce constraints at the expense of robustness. The core problem is designing a control scheme that can adaptively estimate and compensate for state-dependent uncertainties in real-time while simultaneously ensuring the robot’s states remain within safe, time-varying bounds. Achieving this dual objective in a unified, provably stable manner for nonlinear Euler-Lagrange systems is a significant technical hurdle.

The key innovation of this work is the seamless integration of artificial time delay estimation with barrier Lyapunov functions within a unified adaptive control framework. The approach introduces a structured, state-dependent upper bound for the TDE approximation error, which is analytically derived and used to design an adaptive law that updates parameters online. This enables real-time compensation of unknown, state-dependent uncertainties without prior model knowledge. Concurrently, the BLF is employed to encode time-varying safety constraints directly into the control law, ensuring the robot’s position and velocity remain within prescribed bounds during transient and steady states. The Lyapunov-based stability proof guarantees asymptotic convergence of errors, robustness against disturbances, and constraint satisfaction. This dual mechanism addresses the limitations of existing methods that handle either uncertainties or constraints separately, providing a comprehensive solution for complex, safety-critical robotic control.

• �� Model the robot dynamics using Euler-Lagrange equations, defining states (positions and velocities) and control objectives.

• �� Implement artificial time delay (TDE) by collecting past state and input data, forming an approximation of unknown dynamics.

• �� Derive a structured, state-dependent upper bound for TDE errors through analytical analysis, capturing the influence of uncertainties.

• �� Design adaptive laws for parameters based on the error upper bound, updating online to compensate for dynamic uncertainties.

• �� Construct a barrier Lyapunov function (BLF) that encodes time-varying safety constraints on position and velocity, ensuring errors stay within bounds.

• �� Formulate the control input by combining the adaptive compensation term with the BLF-based constraint enforcement, ensuring stability.

• �� Conduct Lyapunov stability analysis to prove that the overall system errors are bounded and converge asymptotically.

• �� Validate the control scheme through simulations and experiments on a 5-DoF robotic arm, assessing tracking accuracy, constraint adherence, and robustness under disturbances.

The experimental platform involved a UFactory xArm-5 robotic manipulator equipped with a custom end-effector and NVIDIA Jetson AGX Xavier for real-time computation. The task involved drawing concentric semicircles and erasing the middle one within a confined space, simulating a delicate operation requiring strict position and velocity constraints. The control algorithms were tested under conditions of external disturbances and model uncertainties, with parameters tuned according to the proposed design guidelines. Baseline controllers included an adaptive BLF-based controller (ABLF) and an adaptive TDE controller (ATDC). Performance metrics focused on tracking errors, constraint violations, and robustness under disturbances. Data collected included joint angles, velocities, and error trajectories, analyzed statistically and visually to compare the effectiveness of the proposed method versus baselines. The experiments demonstrated that the proposed controller maintained errors within bounds and completed tasks accurately, outperforming the baselines in safety and precision.

Quantitative analysis revealed that the proposed controller achieved an average position error of 0.64°, significantly lower than ABLF (1.06°) and ATDC (2.31°). Velocity errors were similarly reduced to 2.40°/s, compared to 3.88°/s and 4.06°/s for the baselines. The errors remained within the prescribed bounds throughout the tasks, confirming effective constraint enforcement. Trajectory plots showed the robot accurately followed the desired paths without violations, even under external disturbances. The stability analysis was supported by error convergence plots, illustrating the errors’ asymptotic decay. The robustness was validated by introducing varying disturbance levels, with the proposed method maintaining performance while baselines exhibited deviations or constraint breaches. These results confirm the framework’s capability to handle real-time uncertainties and safety constraints simultaneously.

This control framework is immediately applicable to industrial robotic arms engaged in precision assembly, delicate machining, and human-robot collaboration, where safety and accuracy are critical. Its model-free nature allows deployment in environments with uncertain or changing dynamics, reducing reliance on precise modeling. Long-term, the approach can be extended to autonomous vehicles, space robots, and medical devices, where safety constraints are paramount. The ability to handle state-dependent uncertainties and time-varying constraints simultaneously opens new avenues for designing intelligent, adaptive robotic systems capable of operating reliably in unpredictable environments, thus broadening the scope of autonomous applications across industries.

Despite its advantages, the method may face challenges under extreme dynamic conditions, such as rapid disturbances or highly nonlinear behaviors, where error bounds may be underestimated, affecting robustness. Parameter tuning remains a critical step; improper initialization can slow convergence or cause instability. Computational complexity increases with system size, potentially limiting real-time performance in high-DoF or highly nonlinear systems. Future work should focus on reducing computational load, enhancing adaptivity, and extending the framework to underactuated and multi-agent systems. Additionally, integrating machine learning techniques could improve uncertainty estimation and control performance in highly uncertain or unstructured environments.