Truncated Taylor Control Barrier Function

• TTCBF is a safety framework that uses truncated Taylor series to approximate barrier functions for enforcing forward invariance in sampled-data control-affine systems.
• It reduces tuning complexity compared to high-order CBF methods by relying on a single class K function and a computed bound on the Taylor remainder.
• TTCBF has been validated in applications like robotic navigation and quadrotor control, demonstrating practical, robust, and real-time safety filtering.

The Truncated Taylor Control Barrier Function (TTCBF) is a framework for enforcing safety in sampled-data, control-affine nonlinear systems, particularly in the presence of high-relative-degree safety constraints. TTCBFs employ Taylor series expansions of barrier functions, truncated at the relative degree, to synthesize control inputs that guarantee forward invariance of safe sets in discrete-time or zero-order-hold implementations. By integrating a single class $\mathcal{K}$ function and bounding the Taylor remainder, TTCBFs generalize Control Barrier Functions (CBFs) to arbitrary relative degree, offering a reduction in tuning complexity compared to high-order CBF methods and enabling practical, robust, real-time safety filtering in applications ranging from robotic navigation to real quadrotor systems (Xu et al., 21 Jan 2026, Xu et al., 19 Mar 2025, Liu et al., 14 Nov 2025, Xiao et al., 12 Dec 2025, Zhang et al., 2021).

1. System Model and Safety Problem

Consider a control-affine nonlinear system,

$$\dot x = f(x) + g(x) u, \qquad x\in\mathbb{R}^n,~ u \in U \subset \mathbb{R}^m,$$

where $f$ and $g$ are sufficiently smooth vector fields. The control input $u$ is typically held constant within each sampling interval under a zero-order hold (ZOH) implementation: $u(t) = u_k$ for $t\in[t_k, t_{k+1})$.

A safety requirement is encoded by a $C^{r+1}$ barrier function $h:\mathbb{R}^n \to \mathbb{R}$, with safe set $\mathcal{C} = \{x~:~h(x)\ge0\}$; here, $r$ is the relative degree of $h$, i.e., the minimal $r$ for which control $u$ appears in $d^r h(x(t))/dt^r$.

The safety-critical control problem is to design a feedback input $u_k$ such that $h(x(t)) \ge 0$ for all $t\ge 0$ given $h(x(0))\ge 0$.

2. Principle of TTCBF Construction

2.1 Taylor Series Expansion and Discrete-Time Approximation

The TTCBF approach centers on the truncated Taylor expansion of the barrier function along the system trajectory, evaluated over a sampling period $T_s$. For a relative degree $r$ barrier, Taylor's theorem yields

$$h(x(t_k + T_s)) = h(x_k) + \sum_{i=1}^{r} \frac{T_s^i}{i!} h^{(i)}(x_k, u_k) + R_r,$$

where $h^{(i)}$ is the $i$-th time derivative of $h$ along closed-loop trajectories, and $R_r$ is the Lagrange remainder of order $r+1$, which depends on higher-order derivatives evaluated at an intermediate point $\xi\in[t_k, t_k+T_s]$.

2.2 TTCBF Inequality

A single class $\mathcal{K}$ function $\alpha$ is used to induce decay in the safety function, leading to the TTCBF discrete-time safety condition: $$\sum_{i=1}^{r} \frac{T_s^i}{i!} h^{(i)}(x_k, u_k) + \alpha(h(x_k)) + U(h, x_k, T_s) \ge 0,$$ where $U(h, x_k, T_s)$ is a computable, typically conservative upper bound on the Taylor remainder $R_r$: $U(h, x_k, T_s) = \frac{M_k}{(r+1)!} T_s^{r+1},$ with $M_k$ a uniform upper bound on $|d^{r+1} h(x(t))/dt^{r+1}|$ in a neighborhood of $x_k$ (Liu et al., 14 Nov 2025, Xu et al., 19 Mar 2025).

This condition is affine in $u_k$ and can be directly implemented as a constraint in a quadratic program (QP), ensuring forward invariance of the discrete safe set under sampled-data control (Xu et al., 21 Jan 2026, Xiao et al., 12 Dec 2025, Zhang et al., 2021).

3. Theoretical Properties and Comparison to High-Order CBFs

TTCBFs generalize standard CBFs to any relative degree without introducing nested auxiliary barrier functions or multiple class $\mathcal{K}$ functions:

• High-Order CBFs (HOCBFs): For relative degree $r$, require a chain of auxiliary functions (e.g., $\psi_0=h$, $\psi_1 = \dot h + \alpha_1(h)$, etc.) and $r$ independently tuned class $\mathcal{K}$ functions, resulting in increased parameterization and tuning complexity, along with a chain of forward invariance conditions on nested sets.
• TTCBFs: Require only a single class $\mathcal{K}$ function, with all conservatism or aggressiveness determined by its gain. The burden of high-dimensional parameter search is avoided; design complexity is reduced from $\mathbb R^r$ to $\mathbb R^1$ (Xu et al., 19 Mar 2025, Xu et al., 21 Jan 2026).

The TTCBF condition is both necessary and sufficient for safety on the sampling interval in the local, linearized tube around $x_k$, provided the remainder is tightly bounded (Xiao et al., 12 Dec 2025).

4. Implementation Methodology

The TTCBF framework is typically realized via the following procedure per sampling step (Zhang et al., 2021, Liu et al., 14 Nov 2025):

1. State Measurement: Acquire the current state $x_k$ (or a robustified estimate $\hat{x}_k$ in case of measurement uncertainty).
2. Taylor Model Computation: Form the truncated Taylor series of the barrier along trajectories starting from $x_k$, up to order $r$.
3. Remainder Bounding: Compute or estimate the remainder bound $U(h, x_k, T_s)$ using local Lipschitz, Hessian, or polynomial bounds.
4. QP Formulation: Construct the safety constraint as a linear inequality in $u_k$ and embed it in a quadratic program:

$$\begin{aligned} \min_{u_k \in U} \quad & \|u_k - u_{\text{nom}}\|^2 \ \text{subject to} \quad & A_k\, u_k \geq b_k \end{aligned}$$

where $A_k$ and $b_k$ are derived from the Lie derivatives and remainder bound.

1. Input Application: Apply $u_k^*$ as a constant input over $[t_k, t_{k+1})$.
2. Robustification (if required): In presence of measurement and actuation uncertainties, the state tube and input set are tightened accordingly; interval Taylor models and reachable set overapproximations are employed (Zhang et al., 2021).

The framework admits further refinement via event-triggered control, where the QP is re-solved if trajectories exit a predetermined state-tube before the intended sampling time (Xiao et al., 12 Dec 2025).

5. Adaptive TTCBF (aTTCBF) and Enhanced Variants

Adaptive TTCBF (aTTCBF) introduces online adaptation of the class $\mathcal{K}$ gain (Xu et al., 21 Jan 2026): $$\sum_{i=1}^{r} \frac{T_s^i}{i!} h^{(i)}(x_k, u_k) + \eta_k \hat{\alpha}(h(x_k)) + U(h, x_k, T_s) \ge 0,$$ where $\eta_k \in [0,1]$ is optimized at each time step within the QP, allowing the system to automatically scale conservatism for feasibility.

Compared to High-Order Adaptive CBFs (PACBF, RACBF), aTTCBF dramatically reduces the number of tuning parameters: for the corridor navigation benchmark, aTTCBF used 18 versus 72 (PACBF) and 126 (RACBF), while delivering preferable tracking and QP solve times (0.71 ms vs 1.56 ms) without increasing control effort (Xu et al., 21 Jan 2026).

6. Practical Examples and Experimental Validation

High-Order System: In a relative-degree-6 spring-mass system, a TTCBF-based controller guarantees $x_3 \leq x_{3,\rm safe}=3.5$ with tight adherence to the safety specification, while a nominal controller overshoots by 0.4 m and violates safety. The control remains aggressive until near the safety boundary, where it smoothly regulates action (Xu et al., 21 Jan 2026).

Mobile Robot Navigation: For a corridor navigation problem with obstacles modeled as high-relative-degree CBFs ($r=2$), TTCBF and aTTCBF filters maintain strict safety, minimizing both path-tracking error and control effort. The one-dimensional tuning of $\alpha$ stands in contrast with the high-dimensional parametric tuning needed for HOCBF implementations (Xu et al., 19 Mar 2025, Xu et al., 21 Jan 2026).

Real-Time Hardware: On the Crazyflie quadcopter (6-state double integrator, 50 Hz control), TTCBF algorithms (reachable set overapproximation, interval Taylor, Lasserre SDP for margin, final QP) executed in $5$–$8$ ms per sample, demonstrating practical real-time deployability (Zhang et al., 2021).

7. Robustification and Sampling Effects

TTCBFs are explicitly designed for sampled-data system safety. By bounding the Taylor remainder and formulating the safety constraint over the sampling period, TTCBFs ensure forward invariance of the safe set even in the presence of inter-sample growth, measurement, and actuation uncertainties (Zhang et al., 2021, Liu et al., 14 Nov 2025).

Robust implementation involves:

• Overapproximating the reachable tube using interval arithmetic and zonotopes.
• Shrinking input sets and recomputing Taylor bounds under input uncertainty.
• Accommodating uncertainty via tightened constraints in the QP.

This ensures the system remains continuously safe under zero-order-hold, not just at sample points, addressing a key limitation of traditional CBF and HOCBF approaches in sampled-data settings (Liu et al., 14 Nov 2025).

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Summary Table: Comparison of TTCBF and HOCBF Methods

Feature	HOCBF	TTCBF
Relative Degree	Arbitrary, but requires $r$ auxiliary functions	Arbitrary, Taylors to order $r$
Class $\mathcal{K}$	$r$ functions, high-dimensional tuning	Single function, 1-dimensional tuning
Safety Guarantee	Via nested auxiliary sets	Directly on $h(x)\ge0$
QP Implementation	Multiple constraints	Single affine constraint
Discrete-Time Safety	Continuity not explicit under ZOH	Explicitly enforces intersample safety

TTCBF thus provides a scalable, computationally tractable framework for high-order, sampled-data safety-critical control, unifying theoretical guarantees with practical real-time applicability (Xu et al., 21 Jan 2026, Xu et al., 19 Mar 2025, Zhang et al., 2021, Liu et al., 14 Nov 2025, Xiao et al., 12 Dec 2025).