No Undesirable Equilibria, No Collisions: Combining Modulation and Control Barrier Functions for Liveness and Safety in Dynamic Collision Avoidance

Yifan Xue; Nadia Figueroa

cs.ROFeb 20, 2025

No Undesirable Equilibria, No Collisions: Combining Modulation and Control Barrier Functions for Liveness and Safety in Dynamic Collision Avoidance

Authors: Yifan Xue, Nadia Figueroa

Organizations: the Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 USA

Abstract

Control Barrier Function Quadratic Programs (CBF-QPs) provide formal safety guarantees for general control-affine systems and accommodate actuation constraints, but do not inherently guarantee liveness, i.e., sustained progress toward and convergence to a goal. They may admit undesirable equilibria or dysfunctional circulation that impede goal convergence. In contrast, Modulation of Dynamical Systems (Mod-DS) promotes liveness through geometric reshaping of nominal vector fields, but is primarily suited to fully actuated systems and does not directly accommodate general actuation constraints. We establish theoretical connections between these paradigms, showing that normal Mod-DS is a special case of CBF-QP and that reference Mod-DS is linked to CBF-QP through a shared formulation. These connections motivate the Modulated CBF-QP (MCBF-QP) framework, which incorporates geometric guidance from Mod-DS into CBF-QPs to enhance liveness while retaining formal safety and control-affine generality. We develop reference MCBF-QP, which reduces undesirable equilibria and dysfunctional circulation for star-shaped obstacles, and on-manifold MCBF-QP, which eliminates undesirable equilibria and reduces dysfunctional circulation for arbitrarily shaped concave obstacles. Although no Lyapunov function is explicitly constructed, we prove that the tangent guidance constraint introduced into the CBF-QP formulation implicitly enforces the decrease condition of a hybrid Lyapunov function, yielding asymptotic goal convergence and thus liveness under the stated assumptions. Extensive simulation and hardware experiments demonstrate safe and live obstacle avoidance with the proposed MCBF-QP framework in static and dynamic environments.

Figures & tables

Panel of Figure 1 — Figure 1: Real-world (a-c) and simulated experiments in Gazebo (d) with the proposed MCBF-QP strategy (Section 6 ) for undesirable-equilibrium-free navigation in complex and dynamic environments.

Panel of Figure 2 — Figure 2: Isolines displaying (a) $h_{\text{conv}}$ , (b) $h_{\text{star}}$ , and (c) $h_{\text{nstar}}$ in the proximity of respectively a circle, a funnel, and a C-shaped obstacle.

Property	$D(x,x_{o})$	$d(x,x_{o})$	Obstacle Type	# Equilibria
Normal ( Khansari-Zadeh and Billard, 2012 )	$\lambda^{}(x,x_{o})=1-\frac{1}{h(x,x_{o})+1}$ , $\lambda_{e}^{}(x,x_{o})=1+\frac{1}{h(x,x_{o})+1}$	$n(x,x_{o})$	convex	1
Reference ( Huber et al., 2019 )	$\lambda^{}(x,x_{o})=1-\frac{1}{h(x,x_{o})+1}$ , $\lambda_{e}^{}(x,x_{o})=1+\frac{1}{h(x,x_{o})+1}$	$r(x,x_{o})$	convex, star	1
On-Manifold ( Fourie et al., 2024 )	$D_{\nabla}(x,x_{o})+D_{\phi}(x,x_{o})+\varphi_{h}(x,x_{o})D_{h}(x,x_{o})$	$n(x,x_{o})$	all ✓	0 ✓

Table 1: Properties of the three main Mod-DS variants, detailing their diagonal scaling matrix

D(x,x_{o})

, basis matrix

E(x,x_{o})

, and the obstacle types they can traverse without generating trapping regions in the modified DS. The table also lists the number of undesirable equilibria induced per obstacle for the navigable obstacle types.

Figure 3: Trajectories from 10 initial locations toward a target at the origin generated by CBF-QP (columns a-b) and Mod-DS (column c-e) obstacle avoidance methods around 1) a convex, 2) a star-shaped, and 3) a non-star-shaped obstacle. The nominal controller is $\dot{x}_{\text{nom}}=u_{\text{nom}}=-\frac{1}{\|x\|_{2}}x$ . Colors along the trajectories indicate the robot speed magnitude at each state $x$ .

Figure 4: Performance of reference MCBF-QP in single obstacle avoidance with no robot input constraints (first row, (a): full view, (b): close view), in single obstacle avoidance with robot input constraints of $\|u_{\text{mcbf}}\|_{2}\leq 2$ (second row, (a): full view, (b): close view), and in multi-obstacle avoidance respectively with and without robot input constraints of $\|u_{\text{mcbf}}\|_{2}\leq 2$ (third row (a),(b)).

Panel of Figure 5 — Figure 5: Figures (a) - (c) show the top, side, and diagonal views of the unsafe regions defined by the augmented barrier function $h_{\text{aug}}$ in Eq. ( 64 ). Figures (d) - (f) illustrate the same views with the geodesic approximation method applied: the selected circumventing trajectory is shown in red, leading efficiently toward the target location (green ’X’) despite the non-Euclidean nature of the state space.

Figure 6: Standard CBF-QP induces undesirable equilibrium behavior in single non-star-shaped obstacle avoidance (left: full view; right: close-up view).

Figure 7: Performance of on-manifold MCBF-QP in single star-shaped (first row) and non-star-shaped (second row, (a): full view, (b): close view) obstacle avoidance with no robot input constraints, in multi-concave obstacle avoidance(third row, (a): without input constraints, (b): with robot input constraints of $\|u_{\text{mcbf}}\|_{2}\leq 2$ ), given $\nu_{\phi}=1$ . Lastly, pictures on the fourth row show the effects of $\nu_{\phi}$ sizes on the resulted safe trajectories ((a): $\nu_{\phi}=0.1$ , (b): $\nu_{\phi}=10$ ).

Figure 8: Reference point shift along the robot’s heading in the shifted unicycle model.

Panel of Figure 9 — Figure 9: Simulation environment and controller setup for robot comparison tests. Figure (a) presents the five test scenarios, while (b) depicts the robot’s perceived view of the hospital environment.

Panel of Figure 10 — Figure 10: Spider plot summarizing controller performance in hospital navigation at 20 Hz. Values are normalized between the center and outermost layer, where a larger radial extent denotes better performance.

Figure 11: Comparison of the proposed MCBF-QP safety filters with CBF-QP safety filters and MPC-CBF solvers for crowd navigation scenarios using sampling-based MPPI planners as nominal controllers.

	Nominal Controller	Safety Filter	Safe %	Reached %	Infeasible #	Duration (s)
Hospital	MPC-DC	None	100	0	55.4	NA
	MPC-DCBF	None	90	0	7.3	NA
	Linear DS	CBF-QP	75	75	16.2	27.3
		CE-CBF	85	90	21.9	37.0
		MCBF (ours)	100	100	0.0	28.3
	Penalty-based MPC	None	100	5	0.0	23.6

Table 2: Performance comparison of the proposed modulated, circulation-embedded, and conventional CBF-QP formulations under different nominal controllers.

Scenario	Methods	Safe %	Reached %	Duration (s)
convex dynamic	S-CBF	100	0	NA
	S-OnM-MCBF	100	100	12.4
	A-CBF	100	0	NA
	A-OnM-MCBF	100	100	14.3
static concave	S-CBF	100	0	NA
	S-OnM-MCBF	100	100	14.9

Table 3: Performances of proposed On-Manifold Modulated CBF-QP methods in comparison to CBF-QP formulations in real-life experiments using Fetch at 20Hz.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 12 — Figure 12: Numerical validation of Theorem 5.2 : the CBF-QP and normal Mod-DS produce identical streamline directions and magnitudes when $\lambda$ and $\lambda_{e}$ are selected according to the derived equivalence conditions in Theorem 5.2 .

Figure 13: Illustration of the geometric interpretation and relationship between $n(x)$ , $e(x)$ and $r(x)$ used for Theorem 5.3 .

Figure 14: Unconstrained reference Mod-DS (first row) vs. speed-constrained (second row, left) and velocity-constrained reference Mod-DS with

U_{\text{ub}}=-U_{\text{lb}}=[2,2]^{\top}

(second row, middle and right). The color codes for images in the left, middle, and right columns respectively represent

\frac{\|u_{\text{unc}}\|_{2}}{u_{\text{ub}}}

\frac{|u_{1}|}{2}

, and

\frac{|u_{2}|}{2}

, given that

u=[u_{1},u_{2}]^{\top}

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