SafeSpace: Aggregating Safe Sets from Backup Control Barrier Functions under Input Constraints

1. Problem Statement

Control Barrier Functions (CBFs) are a standard framework for ensuring safety in control systems by rendering a safe set forward invariant. However, in practical applications with input constraints (e.g., actuator limits), the certified safe operating region derived from a single CBF is often highly conservative.

• The Limitation: Traditional methods often identify multiple small, disjoint safe sets (e.g., around different equilibrium points with distinct stabilizing controllers) but lack a systematic way to unite them.
• The Challenge: Existing approaches to enlarge safe sets face trade-offs:
  • Viability Kernel methods (Hamilton-Jacobi reachability) are accurate but computationally intractable for high-dimensional systems.
  • Backup CBFs offer scalability but typically rely on a single backup controller, leading to conservative approximations.
  • Logical Compositions: Combining multiple CBFs via logical operators (AND/OR) often introduces non-smoothness or feasibility issues in the resulting Quadratic Programs (QPs), making continuous controller synthesis difficult.

The paper addresses the need to aggregate multiple certified safe regions (both explicit and implicit) into a single, larger, certified safe set while maintaining a continuous, optimization-based safety filter.

2. Methodology

The authors propose a unified framework that combines Combinatorial CBFs with Backup CBFs, refined by the introduction of auxiliary variables to ensure feasibility.

A. Generalized Combinatorial CBFs

The paper refines the existing combinatorial CBF framework (which handles logical compositions like unions and intersections of safe sets) by introducing an auxiliary variable $\omega$.

• Mechanism: Instead of requiring a single control input to satisfy all barrier constraints simultaneously (which is often impossible outside the intersection of safe sets), the framework introduces a relaxation term scaled by $\omega$.
• Constraint Formulation: For a set of CBFs $\{h_j\}$, the condition is relaxed to:
  $$\dot{h}_j(x, u) \geq -\alpha(h_j(x)) - \omega \rho(h_j(x) - h(x))$$
  where $h(x)$ is the aggregated barrier function (e.g., the maximum of $h_j$), and $\rho$ is a positive definite function.
• Conjunctive Compatibility: The authors define a condition where the active CBFs (those currently defining the boundary of the safe set) must be compatible. The auxiliary variable $\omega$ automatically scales to restore feasibility for inactive constraints, ensuring the QP remains solvable.

B. Aggregation of Implicit Safe Sets (Backup CBFs)

The framework is extended to Implicit Safe Sets generated by Backup CBFs.

• Implicit Sets: A backup controller $k_b$ renders a base safe set $C$ forward invariant. The "implicit safe set" $C_I$ consists of all states from which the system can reach $C$ safely under $k_b$ within a time horizon $T$.
• Discretization: Since backup trajectories cannot always be solved in closed form, the authors discretize the time horizon to create a finite set of constraints.
• Aggregation: The paper proposes taking the union of multiple such implicit safe sets (each generated by a different backup controller). This results in a nested max-min structure:
  $$h_{agg}(x) = \max_j \left( \min_k \{ \text{trajectory constraints for backup } j \} \right)$$
• Unified Safety Filter: The authors formulate a single CBF-based Quadratic Program (QP) that optimizes the control input $u$ and the auxiliary variable $\omega$ simultaneously. This QP enforces safety for the aggregated set without requiring switching between controllers or hybrid logic.

3. Key Contributions

1. Generalized Combinatorial CBF Framework: Introduction of an auxiliary variable $\omega$ that enables the logical composition of multiple CBFs. This relaxes the strict feasibility requirements of standard combinatorial CBFs, allowing for the union of safe sets under a condition termed conjunctive compatibility.
2. Aggregation of Implicit Safe Sets: Extension of the framework to combine multiple Backup CBFs. This allows the system to leverage multiple distinct backup controllers to create a significantly larger safe operating region than possible with a single backup strategy.
3. Continuous Safety Filter: Proof that the resulting CBF-QP yields a continuous feedback law over the aggregated safe set. This eliminates the need for complex switching logic or hybrid control analysis when transitioning between different safe regions.
4. Theoretical Guarantees: Formal proofs demonstrating that the aggregated set is forward invariant and that the QP is feasible and continuous under the proposed conditions.

4. Results

The approach was validated through simulations on two spacecraft safety problems:

• Safe Attitude Control:

  • Scenario: A satellite must maintain its heat shield within a safe angle relative to the sun while avoiding input saturation.
  • Setup: Five distinct backup sets were created, each stabilizing the satellite to a different equilibrium point.
  • Outcome: The proposed Combinatorial Backup CBF successfully tracked a trajectory circling the boundary of the safe set. In contrast, a single backup set failed to maintain safety while tracking the boundary, and standard CBFs were too conservative to allow the trajectory.

• Safe Spacecraft Station Keeping:

  • Scenario: A satellite orbits an asteroid, avoiding a debris field (inner constraint) and staying within a science-gathering range (outer constraint), subject to strict thrust limits.
  • Setup: The goal was to track a desired elliptical orbit. Four methods were compared: Standard CBF, Standard Backup CBF, Combinatorial CBF, and Combinatorial Backup CBF.
  • Outcome:
    • Standard CBF: Too conservative; the spacecraft could not approach the desired orbit.
    • Standard Backup CBF: Improved tracking but still limited by a single backup controller's reachability.
    • Combinatorial Backup CBF (Proposed): Achieved the largest control invariant safe set. The spacecraft successfully tracked the desired elliptical orbit for a significant portion of the mission while strictly adhering to debris avoidance, science constraints, and input bounds.

5. Significance

This work represents a significant advancement in safety-critical control for systems with actuation limits:

• Operational Envelope Expansion: It provides a principled method to transform multiple small, locally certified safe regions into a single, large, globally certified safe region.
• Scalability: By avoiding the computational complexity of Hamilton-Jacobi reachability and the conservatism of single-controller backup methods, the approach is scalable to complex systems like spacecraft.
• Implementation Simplicity: The method relies on a single continuous QP solver, making it practical for real-time implementation without the need for complex hybrid automata or switching logic.
• Space Applications: The successful demonstration on spacecraft attitude and station-keeping problems highlights its immediate applicability to autonomous space missions where input constraints are severe and safety is paramount.