Probabilistically Safe Controllers for Autonomous Systems Using Scenario-Based Model Predictive Control and Control Barrier Functions

Core Concepts

This paper proposes a safety formulation that combines the strengths of model predictive control (MPC) and control barrier functions (CBFs) to design probabilistically safe controllers for autonomous systems in uncertain environments.

Abstract

The paper presents a safety formulation that solves a finite horizon optimization problem at each time step, like MPC, but enforces probabilistic safety constraints via CBFs only at the first step of the horizon. This approach leverages the strengths of both MPC and CBFs:

1. MPC deals with safety constraints in a direct manner, but its computational demands grow with the prediction horizon length.
2. CBFs are computationally efficient for safety analysis, but can be short-sighted or overly conservative in control invariance calculations.

The proposed method uses a scenario-based approach to transform the probabilistic CBF constraints into a finite number of deterministic CBF constraints. This data-driven method avoids assumptions about the underlying uncertainty's distribution or set geometry.

The authors provide distribution-free, a priori guarantees on the system's closed-loop expected safety violation frequency. They demonstrate the effectiveness of their approach through a case study on unmanned aerial vehicle (UAV) collision-free position swapping and compare it with a state-of-the-art stochastic CBF method.

Stats

The dynamics of the UAVs are affected by an additive disturbance modeled as a normal distribution N(0, 1).
The collision avoidance constraint is encoded using a discrete-time control barrier function with a linearization around the current states.
The acceleration input of each UAV is constrained between -4 m/s^2 and 4 m/s^2.

Quotes

"CBFs provide a principled approach to guarantee safety, while being suitable across various applications, including deterministic robotic systems settings [2], guaranteeing safety of learning methods [23], and providing safety assurances under stochastic conditions [8], which is relevant in uncertain environments."
"While CBFs are computationally efficient for safety analysis, they can be short-sighted or overly conservative in control invariance calculations. To address this, predictive safety filters have emerged as a promising alternative, often used as an "add-on" to existing control strategies [18]."

Key Insights Distilled From

Probabilistically safe controllers based on control barrier functions and scenario model predictive control

by Allan Andre ... at arxiv.org 09-12-2024

https://arxiv.org/pdf/2409.06834.pdf

Deeper Inquiries

How could the proposed approach be extended to handle time-varying or non-linear system dynamics?

The proposed approach, which integrates Control Barrier Functions (CBFs) with scenario-based Model Predictive Control (MPC), can be extended to accommodate time-varying or non-linear system dynamics through several strategies.

Dynamic Linearization: For non-linear systems, one effective method is to employ dynamic linearization techniques. This involves approximating the non-linear dynamics around the current operating point using Taylor series expansion. By linearizing the system at each time step, the existing framework can be applied iteratively, allowing the controller to adapt to changes in system dynamics over time.

Adaptive Control Barrier Functions: The CBFs can be designed to be adaptive, meaning they can adjust their parameters based on the current state of the system. This adaptability can help maintain safety guarantees even as the system dynamics change. For instance, the CBFs could incorporate state-dependent terms that account for the non-linearities in the system.

Time-Varying Constraints: To handle time-varying dynamics, the safety constraints encoded by the CBFs can be modified to reflect the changing environment or system parameters. This could involve updating the CBFs at each time step based on the latest information about the system's state and the external conditions affecting it.

Non-linear Scenario-Based MPC: The scenario-based MPC framework can be adapted to handle non-linear dynamics by utilizing non-linear programming techniques. This may involve reformulating the optimization problem to accommodate non-linear constraints and objectives, potentially using numerical solvers that can handle such complexities.

Robustness to Disturbances: Incorporating robust control techniques can enhance the system's ability to handle uncertainties and disturbances in non-linear dynamics. This could involve designing the CBFs to account for worst-case scenarios or using robust optimization methods within the MPC framework.

By implementing these strategies, the proposed approach can effectively manage the complexities associated with time-varying and non-linear system dynamics while ensuring probabilistic safety guarantees.

What are the potential limitations of the scenario-based approach in terms of scalability and computational complexity as the number of agents or constraints increases?

The scenario-based approach, while powerful, does face several limitations regarding scalability and computational complexity, particularly as the number of agents or constraints increases:

Increased Computational Load: As the number of agents increases, the scenario-based MPC must generate and evaluate a larger number of scenarios to maintain safety guarantees. This leads to a combinatorial explosion in the number of possible state trajectories that need to be considered, significantly increasing the computational burden.

Complexity of Constraint Handling: With more agents, the number of safety constraints derived from CBFs also increases. Each agent's constraints must be satisfied simultaneously, which can complicate the optimization problem. The resulting optimization problem may become non-convex and harder to solve, requiring more sophisticated and time-consuming numerical methods.

Real-Time Implementation Challenges: The need for real-time computation in safety-critical applications means that the scenario-based approach must be able to solve the optimization problem within a limited time frame. As the number of agents and constraints grows, meeting these real-time requirements becomes increasingly challenging, potentially leading to delays or failures in control actions.

Scalability of Scenario Generation: The process of generating scenarios must also scale with the number of agents. If the scenarios are drawn from a distribution, ensuring that they adequately represent the underlying uncertainties becomes more complex. This may require more sophisticated sampling techniques, which can further increase computational demands.

Memory and Storage Requirements: Storing and processing the data associated with a large number of agents and scenarios can lead to significant memory usage. This can be a limiting factor in embedded systems or platforms with constrained resources.

To address these limitations, strategies such as scenario reduction techniques, parallel computing, and hierarchical control architectures could be explored to enhance the scalability and efficiency of the scenario-based approach.

How could the method be adapted to incorporate additional objectives, such as energy efficiency or mission completion, while maintaining probabilistic safety guarantees?

Incorporating additional objectives, such as energy efficiency or mission completion, into the scenario-based MPC framework while maintaining probabilistic safety guarantees can be achieved through several adaptations:

Multi-Objective Optimization: The optimization problem can be reformulated as a multi-objective optimization problem, where the objectives of safety, energy efficiency, and mission completion are balanced. This can be done by introducing a weighted sum of the objectives or using Pareto optimization techniques to find a set of optimal solutions that trade off between conflicting objectives.

Energy-Aware Control Barrier Functions: The CBFs can be designed to include terms that account for energy consumption. For instance, the control inputs can be penalized based on their energy usage, encouraging the controller to select actions that minimize energy expenditure while still satisfying safety constraints.

Mission-Specific Constraints: Additional constraints can be integrated into the CBF framework that reflect mission-specific requirements. For example, constraints could be added to ensure that the agents reach their target positions within a certain time frame or that they follow a specific trajectory that optimizes mission completion.

Adaptive Weighting of Objectives: The weights assigned to different objectives can be made adaptive based on the current state of the system or the environment. For instance, during critical phases of a mission, safety might be prioritized, while in less critical phases, energy efficiency could be emphasized.

Hierarchical Control Structure: A hierarchical control structure can be implemented, where a high-level planner determines the mission objectives and a low-level controller ensures safety and energy efficiency. This separation allows for more complex mission planning while still adhering to safety constraints.

Probabilistic Guarantees for Additional Objectives: The framework can be extended to provide probabilistic guarantees not only for safety but also for the additional objectives. This could involve deriving new theoretical results that characterize the expected performance regarding energy consumption or mission completion times under the scenario-based approach.

By integrating these adaptations, the scenario-based MPC framework can effectively balance multiple objectives while ensuring that safety remains a top priority, thus enhancing the overall performance of autonomous systems in complex environments.