High Order Control Barrier Functions

• High Order Control Barrier Functions (HOCBFs) are a formal generalization of control barrier functions that enforce safety in systems with relative degree greater than one through recursive conditions.
• They employ recursive differential or difference inequalities embedded in convex optimization and MPC frameworks to guarantee the forward invariance of defined safe sets.
• Empirical results across robotics, autonomous vehicles, and multi-agent systems demonstrate HOCBFs’ effectiveness in providing robust, real-time safety with efficient computational performance.

High Order Control Barrier Functions (HOCBFs) are a formal generalization of control barrier functions for safety-critical systems whose safety constraints possess relative degree greater than one. These functions prescribe mathematically rigorous feedback conditions—recursive difference or differential inequalities—for ensuring the forward invariance of “safe sets” defined by zero-superlevel conditions on constraint functions. HOCBFs have enabled robust, real-time safety enforcement for nonlinear and sampled-data systems, MPC frameworks, robotics, autonomous vehicles, and beyond, with ongoing advances in adaptivity, learning, and convex optimization methodology.

1. Mathematical Foundations: Recursive Definition and Invariance

Consider a control-affine system $\dot{x} = f(x) + g(x)u$, $x \in \R^n$, $u \in \U \subset \R^m$, and a constraint $h(x)$ whose zero-superlevel set $\C = \{x : h(x) \ge 0\}$ is the candidate safe set. If $h$ has relative degree $r$—i.e., the control input $u$ first appears in the $r$-th time derivative of $h$—the standard CBF formulation is insufficient. HOCBFs resolve this for $r > 1$ via a sequence of auxiliary functions (class-$\K$ cascade): $$\psi_0(x) := h(x),\qquad \psi_i(x) := \dot{\psi}_{i-1}(x) + \alpha_i(\psi_{i-1}(x)),\quad i = 1,\ldots,r$$ with each $\alpha_i$ a strictly increasing class-$\K$ function, commonly linear or polynomial.

The core HOCBF condition is: $\sup_{u \in \U}\left[ L_f^r h(x) + L_g L_f^{r-1}h(x) u + \sum_{i=1}^{r-1} L_f^{r-i} \bigl[\alpha_i(\psi_{i-1}(x))\bigr] + \alpha_r(\psi_{r-1}(x)) \right] \ge 0$ for all $x$ in the intersection $\bigcap_{i=1}^r \{x:\psi_{i-1}(x)\ge0\}$.

For discrete-time systems $x_{t+1} = l(x_t, u_t)$, the Lie-derivative conditions are replaced by forward-difference inequalities: $$\psi_0(x_t) := h(x_t), \qquad \psi_i(x_t) = \psi_{i-1}(x_{t+1}) - \psi_{i-1}(x_t) + \gamma_i \psi_{i-1}(x_t)$$ with $0 < \gamma_i \leq 1$ regulating decay, and the discrete invariance condition

$$\psi_{r-1}(x_{t+1}) \geq (1 - \gamma_r)\psi_{r-1}(x_t)$$

(Liu et al., 2022, Panja, 17 Aug 2024, Wei et al., 10 Mar 2025, Xiao et al., 2019).

The forward invariance theorem guarantees that if $x_0$ lies in the intersection of the intermediate safe sets and $u_t$ satisfies the HOCBF inequality at each step, then the full chain of sets remains invariant—implying $h(x_t) \ge 0$ for all $t$.

2. Controller Synthesis: Convex Optimization and Model Predictive Control

HOCBF constraints are typically enforced via quadratic programming (QP), convex optimization, or second-order cone programming (SOCP) (Liu et al., 2022, Aali et al., 14 Mar 2024, Wei et al., 10 Mar 2025). For continuous or sampled-data MPC frameworks, the optimization structure is:

$$\min_{U, \Omega} \quad p(x_{t+N}) + \sum_{k=0}^{N-1} q(x_{t+k}, u_{t+k}, \omega_{t+k})$$

subject to system dynamics, input constraints, and HOCBF constraints for all orders $i$, and time steps $k$: $$\psi_{i-1}(x_{t+k}) + \sum_{v=1}^i Z_{v,i} (1-\gamma_i)^k \psi_0(x_{t+v}) \geq \omega_{i,t+k} Z_{0,i} (1-\gamma_i)^k \psi_0(x_t)$$ where slack variables $\omega \ge 1$ introduce feasible relaxation in the decay rates.

Nonlinearities in dynamics or HOCBFs can be addressed via iterative linearization (successive convexification), yielding a sequence of convex QPs rapidly solved at each MPC step. Experimental results in high-dimensional obstacle avoidance benchmarks confirm ≥10× speedup over non-convex MPC formulations with no significant loss in feasibility or safety (Liu et al., 2022).

3. Feasibility, Penalty Methods, and Adaptive Extensions

Control input bounds and high relative degree constraints can inherently compromise QP feasibility. Several strategies have been adopted:

• Slack variable introduction, penalizing violation severity in the cost function.
• Cascade-wise gain tuning: choosing lower $\K$-class gains for less aggressive barrier enforcement to expand the feasible set (Xiao et al., 2019).
• Penalty methods, including adaptive (time-varying) penalty parameters, sometimes endowed with CLF-like stabilization to keep their values near design targets (Xiao et al., 2020, Toulkani et al., 12 Nov 2024).
• Parametric Adaptive HOCBFs (PACBFs): time-varying penalization parameters allow always-feasible online adaptation, preserving safe set invariance even under control and actuator constraints (Toulkani et al., 12 Nov 2024).
• Feasibility learning: trained classifiers (e.g., SVM or shallow neural nets) augment the QP with an additional, data-driven HOCBF representing known feasibility regions (Xiao et al., 2023).

4. Safety Guarantees and Robustness

The theoretical guarantee is forward invariance of the intersection of companion sets—assuming feasibility, suitable class-$\K$ choice, and sufficient smoothness—and holds for (a) continuous-time systems via Nagumo’s theorem, (b) discrete-time systems via recursive application of difference inequalities, and (c) systems with unknown or adaptive dynamics when robustified by event-triggered or model-adaptive surrogates (Liu et al., 2022, Xiao et al., 2021). Extensions encompass input-to-state safety (ISSf), in which tunable functions parameterize invariance under bounded external disturbances (Wei et al., 10 Mar 2025).

For sampled-data systems, compensation terms analytically bound the drift over the sample interval, transforming continuous HOCBFs into "sampled-data HOCBFs" (SdHOCBFs); embedding these into SOCP-based MPC guarantees convexity, safety, and optimality (Gao et al., 6 Oct 2025).

5. Advanced Topics: Multi-Input Systems, Non-Affine Dynamics, Learning, and Verification

Systems with $q > 1$ control inputs present channel-dependent relative degree issues addressed by (i) introducing auxiliary integrator states to uniformly elevate all control channels into the barrier constraint, or (ii) applying constraint transformation if system geometry permits a uniform relative degree (Xiao et al., 2022). For non-affine control systems, differentiable HOCBFs can be embedded into neural-ODE architectures, enabling efficient learning of optimal safety-filtered policies and guaranteeing safety even in high-dimensional, observation-driven domains (Xiao et al., 2023).

Recent work in SOS programming provides constructive methods for verifying feasibility and synthesizing class-$\K$ functions in high-order CBFs, producing safety certificates for complex systems with multiple constraints and polynomial vector fields (Pond et al., 5 Feb 2025).

6. Design and Implementation Considerations

Tuning class-$\K$ gains remains critical: higher gains enforce more aggressive safety but may conflict with tracking objectives and reduce feasibility. Analytical insights on the hierarchical flattening of standard HOCBF chains reveal that gain selection is coupled with the initial conditions and can result in highly non-monotonic or coupled exponential lower bounds on the constraint function’s decay—implicating design complexity for high relative degree. Truncated Taylor-based discrete-time CBFs offer parameter reduction, requiring only a single $\K$ function and a truncation margin (Xu et al., 19 Mar 2025).

Computational scaling relates to the evaluation of higher-order derivatives and solving small-scale QPs. Efficient real-time implementations leverage code generation, iterative convexification, and warm starts in convex solvers.

7. Applications and Empirical Results

HOCBFs have been successfully demonstrated in:

• Robust obstacle avoidance in receding-horizon MPC frameworks (Liu et al., 2022, Gao et al., 6 Oct 2025).
• Contact force limitation and whole-body collision avoidance in soft robotic manipulators, including real-time deployment and geometric reasoning via CBFpy and DCSAT (Wong et al., 5 May 2025).
• Safety-critical control in quadrotors, unicycles, legged robots, and autonomous vehicles under high-dimensional or torque-level constraints (Wei et al., 24 Oct 2024, Xiao et al., 2022).
• Overtaking, cruise control, and adaptive safety systems under matched/unmatched disturbance, actuator lag, and geometric nonlinearities (Wei et al., 10 Mar 2025, Toulkani et al., 12 Nov 2024).
• Learning-based safety filtering via Gaussian processes and sample complexity–driven chance constraint convexification (Aali et al., 14 Mar 2024).

Empirical data consistently verify forward-set invariance, feasibility under tight bounds, control effort reduction, and reduction in conservativeness with tunable gain and penalty methods. Experimental platforms include simulated and hardware robots, automotive systems, and multi-agent collision avoidance.

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In summary, High Order Control Barrier Functions constitute a mathematically rigorous paradigm for enforcing safety in nonlinear and high-relative-degree dynamic systems. Recent advances have rendered their synthesis, verification, adaptive extension, and convex embedding tractable, with diverse applications in robotics, autonomous systems, and safety-critical control. Emerging lines of inquiry include robust and stochastic extensions, gain synthesis via SOS or learning, compositional safety for multi-agent systems, and scalability to high-dimensional and non-affine domains. (Liu et al., 2022, Panja, 17 Aug 2024, Wei et al., 10 Mar 2025, Aali et al., 14 Mar 2024, Xiao et al., 2019, Wong et al., 5 May 2025, Liu et al., 30 Mar 2025, Xiao et al., 2021, K et al., 18 Jul 2025, Gao et al., 6 Oct 2025, Toulkani et al., 12 Nov 2024, Xiao et al., 2021, Wei et al., 24 Oct 2024, Xiao et al., 2023, Xiao et al., 2023, Xu et al., 19 Mar 2025, Xiao et al., 2020, Pond et al., 5 Feb 2025, Xiao et al., 2022).