Response-Aware Risk-Constrained Control Barrier Function With Application to Vehicles

Qijun Liao; Jue Yang

math.OCMar 13, 2026

Response-Aware Risk-Constrained Control Barrier Function With Application to Vehicles

Authors: Qijun Liao, Jue Yang

Organizations: School of Mechanical Engineering, University of Science and Technology Beijing, Beijing, 100083, China

Abstract

This paper proposes a unified control framework based on Response-Aware Risk-Constrained Control Barrier Function for dynamic safety boundary control of vehicles. Addressing the problem of physical model parameter mismatch, the framework constructs an uncertainty propagation model that fuses nominal dynamics priors with direct vehicle body responses. Utilizing simplified single-track dynamics to provide a baseline direction for control gradients and covering model deviations through statistical analysis of body response signals, the framework eliminates the dependence on accurate online estimation of road surface adhesion coefficients. By introducing Conditional Value at Risk (CVaR) theory, the framework reformulates traditional deterministic safety constraints into probabilistic constraints on the tail risk of barrier function derivatives. Combined with a Bayesian online learning mechanism based on inverse Wishart priors, it identifies environmental noise covariance in real-time, adaptively tuning safety margins to reduce performance loss under prior parameter mismatch. Finally, based on Control Lyapunov Function (CLF), a unified Second-Order Cone Programming (SOCP) controller is constructed. Theoretical analysis establishes convergence of Sequential Convex Programming to local Karush-Kuhn-Tucker points and provides per-step probabilistic safety bounds. High-fidelity dynamics simulations demonstrate that under extreme conditions, the method not only eliminates the output divergence phenomenon of traditional methods but also achieves Pareto improvement in both safety and tracking performance. For the chosen risk level, the per-step safety violation probability is theoretically bounded by approximately 2%, validated through high-fidelity simulations showing zero boundary violations across all tested scenarios.

Figures & tables

Symbol	Meaning
$\mathbf{r}=[\beta,\omega_{z},a_{y}]^{T}$	Body response vector
$\tilde{\mathbf{r}}$	Measured response with noise
$\mathbf{F}_{z}=[F_{z,1},\ldots,F_{z,n}]^{T}$	Vertical load vector
$\boldsymbol{\Sigma}_{r},\boldsymbol{\Sigma}_{F}$	Covariance matrices of measurement
	noise and load estimation error
$h(\mathbf{r},\mathbf{F}_{z})$	Barrier function

Table 1: Main symbol definitions

Figure 2: Simplified nominal model of 6 wheels distributional vehicle

Parameter	Symbol	Value
Vehicle mass	$m$	45,000 kg
Moment of inertia	$I_{z}$	3,446,811 kg·m²
Front axle distance	$a$	3.155 m
Rear axle distance	$b$	3.155 m
Track width	$t$	4.147 m
Cornering stiffness (single wheel)	$C_{\beta}$	$1.728\times 10^{6}$ N/rad

Table 2: Main vehicle parameters

Parameter	Symbol	Value
Sideslip angle safety limit	$\beta_{\text{lim}}$	0.15 rad (8.6°)
Yaw rate limit	$\omega_{\text{lim}}$	0.20 rad/s (11.5°/s)
Lateral acceleration limit	$a_{y,\text{lim}}$	5.0 m/s²
Roll angle limit	$\phi_{\text{lim}}$	20°
Prediction window	$T_{\text{pred}}$	0.1 s
Load sensitivity exponent	$\gamma_{w}$	0.3

Table 3: R²CBF controller parameters

Scenario	Algorithm	Safety Indicators				Tracking Indicators
Scenario	Algorithm	$\beta_{\max}$ (°)	$\omega_{\max}$ (°/s)	$a_{y,\max}$ (m/s²)	Margin (%)	RMS( $e_{y}$ ) (m)	RMS( $e_{\psi}$ ) (°)	$\rho_{\text{CBF}}$ (%)
Scenario 1	Pure-CLF	64.03	25.61	4.32	28.43	29.74	91.46	0
	Classic CBF	20.51	16.53	2.15	60.73	2.28	7.89	44.93
	Robust-CBF	18.18	26.35	2.82	40.98	19.42	37.41	51.75
	R²CBF	1.09	7.81	1.20	75.70	1.12	5.73	51.75
Scenario 2	Pure-CLF	39.81	15.58	3.12	31.48	58.07	72.76	0

Table 4: Performance comparison results for two scenarios

Panel of Figure 3 — Figure 3: Comparison of safety indicators in DLC

Panel of Figure 4 — Figure 4: Comparison of tracking indicators in DLC

Figure 5: Comparison of safety margin in DLC

Panel of Figure 6 — Figure 6: Comparison of safety indicators in Sine curve

Panel of Figure 7 — Figure 7: Comparison of tracking indicators in Sine curve

Figure 8: Comparison of safety margin in Sine Curve

Variants	Safety Indicators				Tracking Indicators
Variants	$\beta_{\max}$ (°)	$\omega_{\max}$ (°/s)	$a_{y,\max}$ (m/s²)	Margin (%)	$e_{y}$ (m)	$e_{\psi}$ (°)	$\rho_{\text{CBF}}$ (%)
R²CBF+LoadVar	2.17	8.53	2.43	61.17	1.25	5.45	69.38
w/o CVaR	2.41	8.56	2.55	58.31	1.87	5.31	79.87
w/o Bayesian	1.85	10.57	1.55	86.09	86.80	26.66	100
Full R²CBF	2.15	8.49	2.40	61.81	1.21	5.41	71.21

Table 5: Ablation experiment results comparison

Panel of Figure 9 — Figure 9: Safety indicators comparison of four variants

Panel of Figure 10 — Figure 10: Tracking indicators comparison of four variants

Figure 11: Safety margins comparison of four variants

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