Beyond Quadratic Programming: Streamlining Safety Filters for Robotics

Author: Denis Avetisyan

A new approach to control barrier functions bypasses computationally expensive optimization, paving the way for faster and more efficient safety-critical control systems.

The system demonstrably maintains stable output behavior within safety constraints, as evidenced by its performance on the example detailed in Section 5.2 when governed by a control barrier function-based safety filter.

This work presents a closed-form expression for control barrier function-based safety filters, enabling resource-aware implementation for robotics and reinforcement learning applications.

Guaranteeing safety in increasingly complex control systems often demands computationally expensive real-time optimization. This challenge is addressed in ‘Explicit Control Barrier Function-based Safety Filters and their Resource-Aware Computation’ by deriving a closed-form expression for controllers designed using control barrier functions. This allows for a resource-aware implementation that bypasses quadratic programming solvers, significantly reducing computational burden during operation. Could this approach unlock broader applicability of safety-critical control and reinforcement learning in resource-constrained environments?

Prioritizing Safety: A Paradigm Shift in Control Systems

Historically, the design of control systems has largely focused on achieving high performance – speed, accuracy, efficiency – often treating safety as a secondary consideration or assuming it would be addressed through separate, reactive mechanisms. This approach can lead to vulnerabilities, particularly as systems become more complex and operate in unpredictable environments. Traditional techniques, such as proportional-integral-derivative (PID) control, while effective in many scenarios, don’t inherently prevent a system from exceeding safe operating limits. Consequently, a controller optimized for performance might allow actuators to move beyond their physical constraints or cause a robot to collide with its surroundings. This prioritization has prompted a shift towards control methodologies that explicitly account for and enforce safety boundaries, recognizing that a compromised safety margin negates any performance gains.

The imperative of safety assumes critical importance in the design and deployment of modern robotic and autonomous systems. Unlike traditional control schemes which often prioritize performance metrics like speed or efficiency, these advanced systems operate increasingly in close proximity to humans and within complex, unpredictable environments. Consequently, a failure to rigorously guarantee safety – preventing unintended or harmful actions – can have severe repercussions, ranging from operational disruptions to physical harm. This necessitates a fundamental shift in control philosophy, moving beyond simply achieving a desired task to ensuring that task is completed without compromising safety, demanding robust methodologies for hazard identification, risk assessment, and the implementation of fail-safe mechanisms within the system’s core architecture.

Control strategies are increasingly designed not simply to achieve a desired task, but to fundamentally guarantee safe operation throughout that process. This shift demands a move beyond traditional methods that treat safety as an afterthought, instead embedding limitations directly into the control architecture. Such approaches utilize mathematical frameworks – like control barrier functions and reachability analysis – to define a ‘safe set’ of states, ensuring the system remains within these boundaries regardless of disturbances or uncertainties. By explicitly enforcing these constraints during operation, these strategies prevent potentially hazardous behaviors, even if it means sacrificing some degree of optimal performance. This proactive safety focus is particularly crucial in applications where failures could have severe consequences, such as autonomous vehicles, surgical robotics, and aircraft control systems, enabling reliable and predictable behavior in complex and dynamic environments.

Formally Defining Safety: Control Barrier Functions and Quadratic Programming

Control Barrier Functions (CBFs) provide a methodology for formally encoding safety specifications as inequalities that constrain the system’s state and control inputs. These inequalities are constructed such that, if satisfied, guarantee that the system remains within a defined safe set. Specifically, a CBF, denoted as $h(x(t))$, is a continuously differentiable function of the system state $x(t)$ such that $h(x(t)) \ge 0$ implies safety. The time derivative of the CBF, $\dot{h}(x(t))$, is also considered, and must remain non-negative for the system to maintain safety over time. By formulating safety as these inequality constraints, CBFs enable the synthesis of controllers that explicitly prioritize safety alongside performance objectives.

The integration of Control Barrier Function (CBF) derived inequalities into a Quadratic Programming (QP) framework enables the simultaneous optimization of control inputs and the enforcement of safety constraints. Specifically, the CBF inequalities, which represent safety requirements as constraints on the system’s state and input, are incorporated as linear constraints within the QP formulation. The objective function within the QP typically minimizes a cost associated with control effort or tracking error, subject to these CBF constraints and potentially other system limitations. Solving this QP yields optimal control inputs, $u$, that minimize the defined cost while demonstrably satisfying the safety specifications encoded by the CBF inequalities, thereby guaranteeing safe operation of the system.

Real-time implementation of Control Barrier Function (CBF)-based control relies heavily on the computational efficiency of Quadratic Programming (QP) solvers. CBF-based control formulates safety constraints as quadratic inequalities, resulting in a QP problem that must be solved at each time step. Operator Splitting QP (OSQP) is particularly well-suited for this task due to its ability to efficiently solve large-scale QPs with sparse active sets, a common characteristic of many CBF-based control problems. OSQP achieves this efficiency through an alternating direction method of multipliers (ADMM) algorithm and an active-set method, enabling it to meet the timing requirements necessary for safe and reliable control of dynamic systems. The solver’s speed is critical, as delays in solving the QP can compromise safety guarantees by allowing the system to violate the established barrier functions.

Reinforcement learning training trajectories are shown in the environment (left), demonstrating that Algorithm 1 significantly reduces average iteration time compared to quadratic programming solvers, as indicated by the error bars representing 10 trials (right).

Minimizing Computation: Event-Triggered Control Strategies

Event-Triggered Control (ETC) is a control strategy designed to reduce computational and communication demands in control systems. Traditional periodic control updates occur at fixed intervals, regardless of whether a control action is actually needed. In contrast, ETC initiates control updates only when a predefined triggering condition is met, typically based on the magnitude of a specific error or state deviation. This asynchronous update scheme minimizes data transmission and processing by avoiding unnecessary computations, leading to potential savings in energy consumption, bandwidth usage, and computational resources. The triggering condition is formulated to ensure stability and performance are maintained despite the irregular update schedule, effectively balancing resource reduction with control objectives.

Periodic Event-Triggered Control (ETC) simplifies the implementation of ETC strategies by evaluating triggering conditions at fixed, discrete time intervals, denoted as $T_k$, where $k$ is a positive integer. Instead of continuous monitoring, the system checks if a control update is needed only at these predetermined times. This approach contrasts with aperiodic ETC, which relies on event-dependent triggering, and offers reduced complexity in scheduling and computation. The triggering condition typically involves assessing the difference between the current state and a reference value, or evaluating a Lyapunov function to ensure stability; if the condition is met at time $T_k$, the controller updates its output. The period $T_k$ represents a design parameter that balances the frequency of updates with computational savings.

Combining Event-Triggered Control (ETC) with Control Barrier Function (CBF)-based control strategies enables a reduction in computational demand without compromising system safety. Traditional CBF control requires continuous monitoring and calculation of control inputs, which can be computationally expensive, especially for systems with high dimensionality or fast dynamics. ETC addresses this by introducing a triggering mechanism that activates control updates only when a predefined condition, based on the CBF, is violated. This ensures that safety constraints, as defined by the CBF, are maintained while minimizing unnecessary computations. Specifically, the ETC scheduler evaluates if the current state violates the CBF’s safety margin; if it does, the control input is recalculated and applied. Otherwise, the previous input is maintained, effectively reducing the frequency of computationally intensive control calculations and communication overhead.

Achieving Efficiency: Closed-Form Control Through Explicit CBF-QP

Explicit Control Barrier Functions – Quadratic Programming (CBF-QP) offers a pre-computed, closed-form solution to the online quadratic programming problem typically required for safety-critical control. Instead of solving the QP at each time step, this method pre-calculates the optimal control law as a function of the system state. This is achieved by formulating the QP as a piecewise affine function, allowing direct evaluation of the control input based on the current state without iterative optimization. The resulting control law eliminates the computational burden associated with online QP solvers, leading to significant reductions in processing time and enabling real-time implementation on systems with limited computational resources. The closed-form solution is expressed as $u(x) = Kx + b$, where $x$ represents the state, and $K$ and $b$ are pre-computed gain and bias matrices, respectively.

State space partitioning, a core component of explicit Control Barrier Function Quadratic Programming (CBF-QP), divides the system’s operational space into distinct regions. These regions are defined by identifying which constraints are active – that is, binding or limiting the system’s behavior – at each point in the state space. By pre-computing the solution to the QP problem for each region, the need for real-time optimization is eliminated. This region-based approach allows for a lookup-table methodology where, given the system’s current state, the corresponding pre-computed control action for that region is immediately applied. The boundaries between regions are determined by changes in the active constraint set, creating a piecewise-defined control law that ensures safety and stability throughout the state space.

The Linear Independence Constraint Qualification (LICQ) is a necessary condition for the existence of a solution to the constrained optimization problem underlying closed-form control. LICQ stipulates that at each feasible point, the gradients of the active constraint functions must be linearly independent. Mathematically, if $c_i(x) = 0$ are the active constraints at a point $x^$, then the gradients $\nabla c_i(x^)$ for all active $i$ must not be linearly dependent. Failure to satisfy LICQ indicates that the constraints are redundant or poorly defined at that point, potentially leading to an unbounded or non-existent solution to the quadratic program. Consequently, verifying LICQ is a crucial step in ensuring the mathematical validity and feasibility of the closed-form control solution.

Implementation of the explicit Control Barrier Function – Quadratic Program (CBF-QP) solution is further streamlined through the integration of Sample-and-Hold control. This technique involves applying a piecewise-constant control input, determined offline, that remains constant over discrete time intervals. By pre-computing the control actions for each region of the state space defined by the CBF-QP, online computation is minimized. This discretization, combined with the CBF-QP’s guarantee of constraint satisfaction within each region, provides a provably stable system as long as the sampling rate is sufficient to maintain stability, effectively eliminating the need for real-time control calculations between sampling instances.

Benchmarking of the closed-form control approach against quadratic programming (QP) solvers in quadrotor simulations demonstrates a substantial reduction in computational time. Specifically, results indicate a speedup of up to 6.74x is achievable. This performance gain is attributed to the elimination of iterative online optimization inherent in QP-based methods; the closed-form solution directly computes the control input. This reduction in computation is critical for applications with limited onboard processing capabilities or strict real-time constraints, allowing for higher control frequencies and more complex maneuvers.

The Resource-Aware implementation demonstrates comparable execution time to the OSQP solver while maintaining a lower number of active indices throughout the optimization process.

Safe Learning: Integrating Reinforcement Learning and Control Barrier Functions

The convergence of reinforcement learning and control barrier functions presents a powerful strategy for navigating complex systems while upholding safety critical parameters. Traditionally, reinforcement learning agents learn through trial and error, a process that can be hazardous in real-world applications. By integrating control barrier functions, which mathematically define safe states and control inputs, the learning process is constrained within a defined safety envelope. This synergistic approach allows an agent to explore and optimize its behavior-maximizing rewards-without violating predefined safety constraints, effectively guaranteeing stability and preventing undesirable outcomes. The result is a robust learning framework capable of tackling challenging control tasks in environments where safety is paramount, bridging the gap between the expressive power of reinforcement learning and the reliability of established control theory.

Proximal Policy Optimization (PPO) functions as a dependable reinforcement learning algorithm crucial for establishing a baseline, or nominal, policy within complex systems. This algorithm excels at iteratively refining a policy by taking small, safe steps to improve performance, preventing drastic changes that could destabilize the learning process. PPO’s strength lies in its ability to balance exploration – discovering new strategies – with exploitation – maximizing rewards from known strategies. Through careful updates to the policy, guided by a ‘trust region’ constraint, PPO ensures that each new iteration remains relatively close to the previous one, fostering stability and accelerating the learning of optimal behaviors. This makes it particularly well-suited for applications where consistent and predictable performance is paramount, such as robotic control and autonomous navigation, and serves as a solid foundation when integrated with safety mechanisms like Control Barrier Functions.

The synergy between Control Barrier Functions (CBFs) and Reinforcement Learning (RL) presents a powerful framework for robotic systems operating in complex environments. RL algorithms excel at discovering optimal policies for achieving desired tasks, but often lack inherent safety guarantees during the learning process. CBFs, conversely, provide a mathematically rigorous method for ensuring that a system remains within predefined safe boundaries. By integrating these two approaches, robotic agents can learn intricate behaviors while simultaneously adhering to critical safety constraints. This combination allows for exploration of challenging scenarios without compromising operational safety, effectively bridging the gap between performance and reliability. The resulting CBF-RL framework doesn’t simply react to potential dangers; it proactively prevents unsafe states, opening possibilities for autonomous operation in previously inaccessible domains.

The practical application of Control Barrier Function-Reinforcement Learning (CBF-RL) often faces computational bottlenecks when dealing with complex systems and multiple safety constraints. Explicit Quadratic Programming (QP) offers a solution by pre-computing and storing frequently accessed components of the CBF optimization problem. This resource-aware implementation significantly reduces the computational burden during runtime, enabling CBF-RL to operate effectively in scenarios previously considered too demanding. By streamlining the safety verification process, Explicit CBF-QP not only accelerates learning but also improves the robustness of the resulting policies, allowing for the navigation of intricate environments with numerous constraints while maintaining system safety and stability. The pre-computation approach effectively trades off storage space for processing time, making real-time safety guarantees feasible even in resource-limited settings.

Quadrotor simulations demonstrate a significant optimization in computational efficiency through a resource-aware implementation of Control Barrier Function-Reinforcement Learning (CBF-RL). By carefully managing computational load, the number of calls to the computationally intensive Theta function – crucial for verifying safety constraints – was reduced by 112 out of 2000 during training. This represents a substantial decrease in processing demands, enabling the application of CBF-RL to more complex quadrotor maneuvers and environments previously limited by computational bottlenecks. The reduction in function calls not only accelerates the learning process but also contributes to real-time feasibility, paving the way for safe and efficient autonomous flight in challenging scenarios.

The pursuit of efficiency, central to this work on resource-aware computation for safety-critical systems, echoes a fundamental principle of elegant design. The authors skillfully navigate the complexities of control barrier functions, ultimately arriving at a closed-form expression that circumvents computationally expensive quadratic programming. This simplification isn’t merely a technical achievement; it’s an exercise in discerning what truly needs to be present. As Friedrich Nietzsche observed, “There are no facts, only interpretations.” The authors, through rigorous analysis, have offered a potent reinterpretation of safety filter computation, stripping away unnecessary layers to reveal a leaner, more effective core. This focus on essential elements aligns with a philosophy that values clarity over superfluous complexity, enabling robust performance even within constrained resources.

The Road Ahead

The presented work circumvents a common extravagance: the quadratic program. Yet, simplification is rarely completion. The true measure of this approach will not be in the elegance of its equations, but in its robustness when confronted with the inherent messiness of real-world systems. Current formulations, while computationally efficient, remain tethered to assumptions of full state feedback and relatively static environments. The next iteration must address the inevitable imperfections of sensing and the dynamic, unpredictable nature of the world beyond the simulation.

A critical, and often neglected, consideration is the interplay between safety and learning. Reinforcement learning, in its eagerness to explore, frequently brushes against the boundaries of acceptable behavior. While this work provides a mechanism to react to constraint violations, a more profound advance lies in anticipation. Future efforts should focus on integrating these safety filters directly into the learning process, guiding exploration and shaping policies that are intrinsically safe, rather than simply defensively so.

Ultimately, the goal isn’t simply to compute safety, but to cede it. To design systems that are so fundamentally constrained by their architecture that explicit safety filters become redundant. The pursuit of perfect control is a fool’s errand; the art lies in designing systems that fail gracefully, or better yet, avoid failure altogether through inherent limitation. This, then, is the direction-not towards more complex filters, but towards simpler, more constrained designs.

Original article: https://arxiv.org/pdf/2512.10118.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/