Safety-Critical Adaptive Impedance Control via Nonsmooth Control Barrier Functions under State and Input Constraints

Abstract: Safe physical interaction is critical for deploying robotic manipulators in human-robot interaction and contact-rich tasks, where uncertainty, external forces, and actuator limitations can compromise both performance and safety. We propose an online adaptive impedance control framework that enforces joint-state safety while achieving compliant interaction under uncertain dynamics. The approach combines a quadratic-program-based safety filter with a novel composed position-velocity non-smooth control barrier function (NCBF), enabling joint position and velocity constraints to be enforced through a unified relative-degree-one barrier. Unknown dynamics are compensated online using an interval type-2 fuzzy logic system, while actuator torque limits are handled through soft constraints with exact penalty recovery of feasible solutions. A disturbance-observer-enhanced safety mechanism improves robustness against modelling errors and external interaction forces. Using composite Lyapunov analysis, we prove forward invariance of the safe set and the uniform ultimately boundedness of the impedance-tracking error. Simulations on a 7-DOF manipulator with severe parametric uncertainty and external interaction wrenches demonstrate safe constraint satisfaction and robust impedance tracking.

• The paper introduces a control scheme that ensures safe impedance tracking by enforcing joint and actuator constraints under uncertainty.
• It integrates a nonsmooth control barrier function with a disturbance observer and IT2-FLS for robust adaptive compensation.
• The method employs soft-constrained quadratic programming to guarantee feasibility and stability, validated on a 7-DOF manipulator.

Safety-Critical Adaptive Impedance Control via Nonsmooth Control Barrier Functions under State and Input Constraints

Introduction and Background

This paper presents a comprehensive control scheme for safe physical interaction in robotic manipulators, focusing on scenarios where uncertainty, external disturbances, and actuator limitations complicate both performance and safety. The solution is relevant for human–robot interaction (HRI) settings and other contact-rich tasks that require robust enforcement of joint and actuator constraints, accurate adaptation to uncertain system dynamics, and compliance during interaction with unpredictable environments. The proposed methodology addresses deficiencies in conventional impedance control, model-based schemes, and adaptive control architectures by integrating a safety filter based on a novel nonsmooth control barrier function (NCBF), adaptive function approximation by an interval type-2 fuzzy logic system (IT2-FLS), actuator limit enforcement via soft-constrained quadratic programming (QP), and robustification with disturbance observers (DOBs).

Problem Formulation

A general $n$-DOF manipulator described by Euler-Lagrange dynamics (with significant parametric uncertainty and external forces) is considered. The physical constraints include state (joint position and velocity) and input (actuator torque) limitations. The objective is to realize a Lipschitz-continuous control law $u(x, t)$ that, for all time, ensures (i) joint state invariance within a prescribed safe set, (ii) adherence to actuator torque bounds, and (iii) uniformly ultimately bounded (UUB) impedance tracking error with respect to a desired compliance model.

The salient challenge is the simultaneous enforcement of these constraints under severe model uncertainties, unmodeled interaction wrenches, and joint/actuator limitations, all without recourse to offline training or substantially overconservative safety margins.

Methodological Contributions

Unified Nonsmooth Control Barrier Function Formulation

A key contribution is the construction of a composed position–velocity NCBF, which encodes both position and velocity limits in a single, relative-degree-one barrier. Unlike standard higher-relative-degree barrier approaches, this method yields a Lipschitz-continuous safety filter, with the nonsmooth composition addressed via weak set-valued Lie derivatives [7937882]. The NCBF is embedded as a hard constraint in a QP, ensuring invariance of the joint-state safe set.

A robust adaptation of the NCBF incorporates a DOB, compensating for model mismatch and external disturbances, which allows for less conservative barrier design compared to worst-case bounds.

Adaptive Disturbance Compensation via IT2-FLS

Unknown model components are approximated online using IT2-FLS. This function approximator was selected for its analytical interpretability, capability of universal function approximation [159070], and robust handling of modeling error. The adaptive law employs a command-driven reference model, in the spirit of MRAC but hedges for physical constraint enforcement by using the QP-constrained output as the reference input. This design decouples the adaptation signal from constraint-induced effects, thus preventing learning from adapting against the safety filter—a failure mode that otherwise leads to bursting and chattering, as demonstrated with the AWORM baseline.

Soft-Constrained QP Enforcement of Input Constraints

A single QP is formulated in which CBF-based joint-state constraints are maintained as hard constraints, while actuator torque limits are softened via exact penalty terms [kerrigan_soft_2000]. This architecture ensures exact recovery of feasible solutions when hard satisfaction is possible, and minimal constraint violations (with chattering avoidance) otherwise. The QP’s decision variable is joint acceleration, and the solution’s local Lipschitz continuity—essential for forward invariance and uniqueness of closed-loop solutions—is proved under mild regularity conditions [agrawal_reformulations_2025, bonnans_perturbation_2000].

An online feasibility metric monitors the available wrench capacity in all directions, triggering graceful constraint softening as required during in-the-loop operation.

Stability and Safety Guarantees

Safety and stability are established via composite Lyapunov analysis. Rigorous proofs of forward invariance for the safe set and UUB tracking error are provided, incorporating the nonstandard structure of the NCBF and the adaptive/dynamic learning subsystems [glotfelter_nonsmooth_2017, khalil_nonlinear_2002]. Notably, tuneable parameter intervals are analytically characterized, ensuring non-degeneracy of the feasible set—a recurring challenge in practical CBF enforcement [kim_is_2026].

Numerical Results

Extensive simulation studies on a 7-DOF Kinova manipulator validate the method under severe parametric mismatch (up to $70\%$), unmodeled friction, and pulsed interaction wrenches beyond nominal actuator limits. The findings can be summarized as follows:

• Constraint Satisfaction and Robustness: The proposed controller enforces all joint state and, when feasible, actuator input constraints. It robustly tracks the impedance model with low error, outperforming both a nominal impedance controller (NIC) and an AWORM baseline (adaptation driven directly by impedance error).
• Superiority of Reference-Governed Adaptation: Adaptation based on the command-driven reference-model avoids the bursting and chattering observed in conventional adaptive designs (AWORM), especially under active constraints. This decoupling yields smoother torque profiles and eliminates high-frequency oscillation in the presence of constraint activation.
• Graceful Degradation in the Presence of Infeasibility: With actuator torque limits tightened beyond what the interaction wrenches permit, the soft-constrained QP triggers minimal slack activation. Impedance tracking and reference-model learning error remain bounded; the system avoids both constraint violation and adaptation chattering.

Strong numerical results are reported: RMS tracking error remains $O(10^{-2})$ even under $70\%$ model perturbation; the safety filter enforces strict invariance of the joint state set; constraint softening operates only when hard satisfaction becomes physically impossible.

Practical and Theoretical Implications

This work establishes several important implications for safe robot control:

• Constraint-Consistent Learning for Safety-Critical Adaptive Control: The adaptation decoupled from constraint-enforcement presents a general paradigm applicable to constrained learning-based controllers, eliminating the need for ad hoc constraint “hedging” or passivity-inspired arguments [johnson_pseudo-control_2000, califano_passivity-preserving_2023].
• Nonsmooth, Yet Lipschitz, Barrier Function Construction: The NCBF composition provides practitioners with a toolkit for encoding multiple state constraints within a single, computationally tractable barrier, avoiding regularity pathologies of higher-order or discontinuous CBF designs [glotfelter_nonsmooth_2017].
• Robust Constraint Satisfaction under Uncertainty: Incorporating disturbance observers into safety filters enables less conservative robustification, outperforming worst-case expansion methods and ensuring maximal use of the actuator’s physical capabilities [10156095].
• Analytical Feasibility Characterization: The detailed analytic intervals for parameter tuning ensure the controller is not only theoretically sound but also practically implementable, with explicit bridge to real manipulator constraints and actuator capabilities.

Future Directions

The paper indicates that future research will include extending the function approximation to neural networks (potentially further improving adaptation in high-DOF robots), enforcing task-space safety constraints (not just joint space), and experimental validation across diverse manipulator platforms. Theoretical extensions could include passivity-based synthesis, further regularity guarantees via SOCP formulations [agrawal_reformulations_2025], and hybrid architectures leveraging learned models for model-predictive safety filtering.

Conclusion

This study advances the state of safety-critical control in uncertain robotic manipulators by synthesizing an adaptive impedance controller that uniquely combines NCBF-based safety invariance (with DOB robustness), constraint-consistent online learning (via IT2-FLS), and exact-penalty soft-constrained QP enforcement. The result is a controller with strong formal guarantees and robust empirical performance under tight, realistic operating constraints—a significant step toward certifiable, high-performance physical human–robot interaction and safe autonomous manipulation.

---

References

• "Nonsmooth Barrier Functions With Applications to Multi-Robot Systems" [7937882]
• "Fuzzy basis functions, universal approximation, and orthogonal least-squares learning" [159070]
• "Soft Constraints and Exact Penalty Functions in Model Predictive Control" [kerrigan_soft_2000]
• "Reformulations of Quadratic Programs for Lipschitz Continuity" [agrawal_reformulations_2025]
• "Perturbation Analysis of Optimization Problems" [bonnans_perturbation_2000]
• "Nonsmooth Barrier Functions With Applications to Multi-Robot Systems" [glotfelter_nonsmooth_2017]
• "Is Your Safe Controller Actually Safe? A Critical Review of CBF Tautologies and Hidden Assumptions" (Kim, 7 Mar 2026)
• "Disturbance Observer-based Robust Control Barrier Functions" [10156095]
• "Passivity-Preserving Safety-Critical Control Using Control Barrier Functions" [10136379]
• "Stable Adaptive Systems" [narendra_stable_2012]
• "Projection Operator in Adaptive Systems" (Lavretsky et al., 2011)
• "A robust adaptive nonlinear control design" [polycarpou_robust_1996]
• "Nonlinear Systems" [khalil_nonlinear_2002]
• "A Tutorial Survey and Comparison of Impedance Control on Robotic Manipulation" [Song et al., 2019]
• "State-Constrained Online Adaptive Control for Robotic Manipulators" [11303114]
• "Adaptive Deep Neural Network-Based Control Barrier Functions" (Sweatland et al., 2024)
• "Limiting Kinetic Energy Through Control Barrier Functions: Analysis and Experimental Validation" [11030294]