Enforcing Mixed State-Input Constraints with Multiple Backup Control Barrier Functions: A Projection-based Approach

1. Problem Statement

Modern autonomous systems (e.g., robots, vehicles) require safety guarantees that satisfy not only state constraints (e.g., position, velocity limits) but also input constraints (e.g., force, torque limits) and mixed state-input constraints (e.g., power limits depending on both velocity and force).

• The Challenge: Standard Control Barrier Functions (CBFs) ensure state safety but often fail when input constraints are present because the control input required to maintain safety might lie outside the admissible input set.
• Existing Limitations:
  • Integral CBFs: Augment dynamics but can be complex.
  • Zero-order CBFs: Handle input dependence but often result in non-convex constraints.
  • Standard Backup CBFs: Provide formal safety and feasibility guarantees for decoupled state and input constraints but have not been established for mixed (coupled) state-input constraints. Furthermore, traditional backup CBFs often require "saturating" backup controllers to respect input limits across the entire state space, complicating design and computation.

2. Methodology

The authors propose a projection-based framework that generalizes the Backup Control Barrier Function (Backup CBF) method to handle mixed constraints.

A. Core Concept: Projection

The central idea is to project mixed constraints from the state-input space ($\mathbb{R}^n \times \mathbb{R}^m$) onto the state space ($\mathbb{R}^n$) along a specific backup controller ($k_b$).

• Projected Constraint Set ($C_p$): Instead of enforcing $H(x, u) \geq 0$ directly, the method defines a state-only constraint $h(x) = H(x, k_b(x)) \geq 0$.
• Logic: If the system state remains within a specific invariant set derived from $C_p$, and the backup controller is used, the original mixed constraint is satisfied.

B. Handling Multiple Constraints

The framework extends to handle $N$ simultaneous constraints (state, input, and mixed) using a single backup set ($S_b$) and a single backup controller ($k_b$).

• The invariant set $S_I$ is defined as the set of states from which the system can reach the backup set $S_b$ within a time horizon $T$ while satisfying all projected constraints along the trajectory.
• This unifies the handling of decoupled constraints (where $H$ depends only on $x$ or only on $u$) and coupled constraints.

C. Control Synthesis (The Optimization Problem)

The control input $u$ is determined by solving a Quadratic Program (QP) at each time step:
$$\begin{aligned} k(x) = \underset{u}{\text{argmin}} \quad & \|u - k_d(x)\|^2 \\ \text{s.t.} \quad & \dot{h}_j(\phi_b(\theta, x), u) \geq -\alpha_j(h_j(\phi_b(\theta, x))), \quad \forall \theta \in [0, T], \forall j \\ & \dot{h}_b(\phi_b(T, x), u) \geq -\alpha_b(h_b(\phi_b(T, x))) \\ & H_j(x, u) \geq 0, \quad \forall j \end{aligned}$$

• Recursive Feasibility: The QP is guaranteed to be feasible because the backup controller $k_b(x)$ is always a valid solution (by construction).
• Linearity: If the mixed constraints are affine in the input, the resulting QP is convex and computationally efficient.

D. Simplification of Backup Design

A key theoretical contribution is that this projection method eliminates the need for saturating backup controllers.

• In standard methods, $k_b$ must satisfy input constraints everywhere in the state space.
• In this method, $k_b$ only needs to satisfy input constraints within the small backup set $S_b$. This allows for simpler controller designs (e.g., linear feedback) and enables analytical calculation of the backup flow (avoiding numerical integration).

3. Key Contributions

1. Generalization of Backup CBFs: The first framework to formally handle mixed state-input constraints (coupled constraints) using Backup CBFs.
2. Projection-Based Transformation: A novel technique to convert complex mixed constraints into pure state constraints via projection along a backup controller, enabling the use of standard state-space invariance logic.
3. Unified Multi-Constraint Handling: A method to enforce multiple decoupled and coupled constraints simultaneously using a single backup set-controller pair, simplifying the synthesis process.
4. Design Simplification: Demonstrates that the projection approach removes the requirement for saturated backup controllers, potentially allowing for analytical backup flow computation (reducing computational cost).
5. Recursive Feasibility Guarantee: Proves that the proposed control law ensures the system remains within the safe set indefinitely, even under strict input and mixed constraints.

4. Results

The authors validated the method using an inverted pendulum example subject to:

• State Constraint: Angle limits ($|x_1| \leq \phi_{max}$).
• Input Constraint: Torque limits ($u_{min} \leq u \leq u_{max}$).
• Mixed Constraint: Power limits ($P_{min} \leq x_2 u \leq P_{max}$).

Findings:

• Set Analysis: The projected constraint set $C_p$ and the resulting invariant set $S_I$ were computed analytically. The study showed that $S_I$ shrinks as more constraints (input, then power) are added, correctly reflecting the reduced safe operating region.
• Simulation Comparison:
  • Proposed Method (Backup CBF-QP): Successfully maintained all state, input, and power constraints for unstable initial conditions. The forward prediction allowed the controller to intervene early.
  • Baseline Method (High-Order CBF - HOCBF): When applied to the same problem, the HOCBF-QP became infeasible due to the conflict between the immediate need to satisfy mixed constraints and the lack of predictive safety margins. The trajectories violated input and power limits.
• Efficiency: The use of a linearized backup system allowed for analytical computation of the backup flow, avoiding the computational overhead of online ODE integration.

5. Significance

This paper addresses a critical gap in safety-critical control: the inability of existing CBF methods to robustly handle coupled state-input constraints (like power limits) while guaranteeing recursive feasibility.

• Practical Impact: The method is directly applicable to real-world systems where power, energy, or actuator limits depend on the system's current state (e.g., electric vehicles, legged robots, teleoperation).
• Computational Advantage: By removing the need for saturated controllers and enabling analytical backup flows, the approach makes safety-critical control more computationally tractable for real-time implementation.
• Theoretical Advancement: It provides a rigorous mathematical foundation for treating mixed constraints as state constraints via projection, expanding the applicability of the Backup CBF framework to a broader class of nonlinear control problems.