Hybrid Lyapunov and Barrier Function-Based Control with Stabilization Guarantees

1. Problem Statement

The paper addresses a critical limitation in the standard Control Lyapunov Function (CLF) and Control Barrier Function (CBF) framework, which is typically unified via Quadratic Programs (QPs) to achieve simultaneous stabilization and safety.

• The Core Issue: While standard CLF-CBF-QP controllers ensure the forward invariance of a safe set, they often introduce unwanted equilibrium points (deadlocks) on the boundary of the safe set. In these scenarios, the controller relaxes the stabilization objective to maintain safety feasibility, causing the system to converge to a point other than the desired target, thereby preventing task completion.
• Existing Limitations:
  • Potential Fields/Navigation Functions: Prone to local minima and oscillatory behavior.
  • Model Predictive Control (MPC): Computationally expensive for real-time applications and struggles with non-convex constraints.
  • Standard CLF-CBF-QP: Fails to guarantee global asymptotic stabilization in the presence of complex unsafe sets due to induced deadlocks.
  • Previous Hybrid Approaches: Some existing hybrid methods (e.g., using polytopic avoidance) are overly conservative or restricted to specific polytope geometries where induced equilibria happen to align favorably.

Goal: Design a control strategy that guarantees global asymptotic stabilization to a desired equilibrium point while maintaining safety (forward invariance) for systems with bounded convex polytopic unsafe sets, without inducing deadlocks.

2. Methodology

The authors propose a Hybrid CLF-CBF Control Strategy that combines continuous control laws with discrete switching logic. The approach is developed for first-order systems and extended to higher-order strict-feedback systems.

A. Geometric Setup

• Unsafe Set Approximation: The bounded unsafe set $O$ is enclosed by a convex polytope $P$, defined by the intersection of $Q$ half-spaces.
• Safe Set: The safe set $C$ is defined as the complement of $P$ (the union of the safe half-spaces).
• Hybrid State: The controller utilizes an auxiliary state $\xi = (\hat{x}, q)$, where:
  • $\hat{x}$ is the current active target point.
  • $q$ is the index of the active safe half-space.

B. Switching Logic (The "Hybrid" Component)

The core innovation is a switching mechanism that updates the target point and the active half-space to avoid deadlocks.

1. Reference Direction: A reference vector $v$ is defined pointing toward the desired equilibrium $\bar{x}$.
2. Synergy Gap: A parameter $\mu$ ensures the intermediate target point lies sufficiently deep inside the next safe half-space, preventing the system from getting stuck on the boundary.
3. Hysteresis: A hysteresis width $\sigma$ determines when a "jump" (switch) occurs. A switch is triggered when the system state satisfies specific geometric conditions relative to the current and forecasted half-spaces.
4. Target Update:
   • If the next half-space contains the final goal $\bar{x}$, the target is set to $\bar{x}$.
   • Otherwise, a new intermediate target $\hat{x}^+$ is computed on the boundary of the new half-space, shifted along a tangent vector to satisfy the synergy gap condition.
   • This ensures a sequence of targets that monotonically converges to $\bar{x}$.

C. Subproblem Controller Design

For each mode (defined by a specific target $\hat{x}$ and active half-space $q$), a local controller $k_\xi$ is synthesized:

• First-Order Systems: A QP is solved to minimize control effort subject to:
  • CLF Constraint: Drives the state toward $\hat{x}$.
  • CBF Constraint: Keeps the state within the active half-space $q$.
  • Compatibility: The authors prove that for half-spaces, compatible CLF and CBF constraints can always be found, ensuring the QP is feasible and deadlock-free within that mode.
• Higher-Order Systems (Backstepping): The method is extended to strict-feedback systems (e.g., double integrators) using a joint CLF-CBF backstepping approach. This recursively constructs CLFs and CBFs for the full system from the top-level subsystem, ensuring the safety and stability properties hold for the entire dynamics.

3. Key Contributions

1. Global Asymptotic Stabilization with Safety: The paper provides a rigorous proof that the proposed hybrid strategy guarantees convergence to the desired equilibrium point for any bounded convex polytopic avoidance domain, eliminating the deadlock problem inherent in standard CLF-CBF-QP.
2. Systematic Design Framework: Unlike previous hybrid methods that require specific polytope geometries or fixed induced equilibria, this approach dynamically assigns intermediate targets. This makes the design flexible and applicable to any convex polytope enclosing the unsafe set.
3. Extension to Higher-Order Systems: The framework is successfully extended to higher-order strict-feedback systems via a novel joint CLF-CBF backstepping procedure, maintaining the hybrid switching logic.
4. Handling Input Constraints: The paper discusses how to incorporate input bounds into the subproblem QP formulation while maintaining feasibility.

4. Simulation Results

The authors validate the method through several simulation scenarios:

• First-Order Dynamics (Integrator): Trajectories successfully navigate around pentagons, squares, and triangles to reach the target. The simulations demonstrate that varying the synergy gap ($\mu$) adjusts the conservativeness of the path. Crucially, unlike continuous CLF-CBF approaches, the system never gets stuck at the obstacle boundary.
• Comparison with Literature: The proposed method is compared against a similar hybrid approach from prior work [50]. The results show that the prior method fails (deadlocks) for certain target locations and polytope shapes, whereas the proposed method succeeds in all cases.
• Second-Order Dynamics (Double Integrator): The backstepping extension successfully handles acceleration constraints, producing smoother trajectories than the first-order case.
• Unicycle Model: The method is applied to a non-holonomic unicycle model via a coordinate transformation (controlling a point ahead of the robot). The system successfully avoids obstacles and converges to the target region.

5. Significance

This work represents a significant advancement in safe control for nonlinear systems.

• Theoretical Impact: It resolves the long-standing "deadlock" issue in CLF-CBF frameworks by introducing a geometric hybrid switching logic that guarantees global convergence.
• Practical Impact: The method offers a computationally efficient alternative to MPC (solving small QPs rather than mixed-integer programs) while providing formal safety guarantees.
• Versatility: By handling higher-order dynamics and non-holonomic systems, the framework is directly applicable to a wide range of real-world robotic platforms, including autonomous vehicles, legged robots, and manipulators.

In summary, the paper presents a robust, mathematically rigorous, and flexible control architecture that ensures robots can safely navigate complex environments and reach their goals without getting trapped, overcoming the fundamental limitations of previous continuous control strategies.