Backstepping CBFs for Safety-Critical Systems

• Backstepping CBFs are constructive safety synthesis methods that recursively lift low-dimensional safety functions to full-state barriers using virtual controls.
• They tackle high-relative-degree constraints in cascaded, underactuated, or strict-feedback systems by introducing error coordinates and quadratic error penalties.
• The approach extends to robust, infinite-dimensional, and geometric systems, enabling applications in autonomous vehicles, multicopters, and complex robotic setups.

Backstepping Control Barrier Functions (CBFs) are constructive safety-synthesis methods for systems whose safety-relevant variables are not directly actuated, but are instead connected to the input through integrator chains, strict-feedback structures, underactuated couplings, or more general hierarchical dynamics. In these methods, a safety specification is first posed at a “top level,” typically through a function $h_0$ or $\psi$ on the safety-relevant state, and is then recursively lifted to a valid barrier function on the full system by introducing virtual controls, auxiliary barrier variables, or quadratic error penalties. The resulting barrier is then enforced by a safety-critical controller, often through a quadratic program (QP), so that forward invariance of a full-state safe set implies the original top-level safety requirement (Taylor et al., 2022).

1. Conceptual lineage and problem class

Backstepping CBFs arose from the observation that many safety constraints are high-relative-degree constraints: the control input does not appear in the first derivative of the safety function, and direct application of standard CBF inequalities is therefore impossible. The canonical examples are position constraints for double integrators, mechanical systems whose inputs act at the acceleration level, and cascaded or layered systems in which the safety-relevant state is separated from the control by intermediate dynamics (Taylor et al., 2022).

The literature uses the term in several closely related ways. In the strict-feedback setting, CBF backstepping denotes a recursive construction analogous to Lyapunov backstepping, where a top-level safety controller is synthesized first and then propagated through the cascade to obtain a CBF for the full higher-order system (Taylor et al., 2022). In high-relative-degree PDE control, backstepping can instead appear as an auxiliary-barrier construction that reshapes the dynamics of barrier variables, as in the Stefan model with actuator dynamics, where an additional CBF $h_3$ is introduced to handle a relative-degree-two barrier functional (Koga et al., 2021). A different but related line treats backstepping as a “CBF-space” recursion $h_{i+1}=c_i h_i + L_f h_i$, yielding target CBFs of relative degree one for originally high-order constraints (Kim et al., 21 May 2025).

This family of methods has since been extended to non-strict-feedback and mixed-relative-degree dynamics in multicopters (Kim et al., 2023), robust high-relative-degree systems with disturbances (Kim et al., 2024), multi-robot cooperative herding with underactuated evader dynamics (Li et al., 14 Jul 2025), activated or rectified constructions for less conservative safe sets (Gacsi et al., 28 Aug 2025), and geometric mechanical systems on manifolds such as $SO(3)$ (Sa et al., 23 Oct 2025).

2. Recursive construction in finite-dimensional systems

The basic finite-dimensional construction begins with a control-affine or strict-feedback system and a top-level safety function $h_0$ or $\psi$ defined on a reduced state. For a two-layer strict-feedback system, the top-level dynamics are treated as if an intermediate state $\xi$ were directly selectable, and a virtual safe controller $k_0(x)$ is designed so that

$$\nabla h_0(x)\bigl(f_0(x)+g_0(x)k_0(x)\bigr)\ge -\alpha_0(h_0(x)).$$

Backstepping then lifts this design to the full state by defining a composite barrier of the form

$\psi$0

or, for longer cascades,

$\psi$1

The negative quadratic terms shrink the safe set in the directions corresponding to deviations from the virtual safe dynamics; when those deviations vanish, the composite barrier reduces to the original top-level safety function (Taylor et al., 2022).

A closely related relative-degree-two formulation appears in output-based backstepping. If safety is specified by $\psi$2, with $\psi$3 and $\psi$4, then a virtual safe controller $\psi$5 is chosen for the single-integrator model $\psi$6. Standard backstepping then constructs

$\psi$7

This yields a valid CBF for the original system under suitable assumptions on $\psi$8, and the full-state controller can be implemented either in closed form or through a CBF-QP that minimally modifies a desired input $\psi$9 subject to $h_3$0 (Gacsi et al., 28 Aug 2025).

The distinctive feature of backstepping CBFs is therefore not merely the use of high-order derivatives, but the explicit introduction of error coordinates between actual lower-level states and virtual safe controls. Safety is encoded by ensuring that these error coordinates remain sufficiently small that the top-level barrier margin is not exhausted. In the strict-feedback interpretation, this is the safety analogue of Lyapunov backstepping; in the output-relative-degree interpretation, it is a constructive CBF synthesis method for high-order constraints (Taylor et al., 2022).

3. High-relative-degree safety, rectification, and activated variants

A central theme in the literature is the relation between backstepping CBFs, High-Order CBFs (HOCBFs), and more recent rectified constructions. HOCBFs enforce a sequence of inequalities on recursively differentiated functions $h_3$1, but their safe set is an intersection of multiple constraints rather than a single CBF superlevel set. Several papers emphasize that this distinction matters for robustness, CLF–CBF unification, and QP regularity, especially in weak-relative-degree settings where $h_3$2 may vanish on nontrivial subsets (Ong et al., 2024).

Rectified Control Barrier Functions (ReCBFs) address this by constructing a single CBF

$h_3$3

for relative-degree-two constraints, and by generalizing this idea recursively to higher relative degree. The key design principle is activation only where higher-order corrections are needed: when the uncontrolled dynamics already satisfy a CBF-like inequality, the rectifier deactivates and $h_3$4. The same paper explicitly compares this to backstepping CBFs, noting that backstepping can naturally handle mixed-input relative degree through recursive virtual-control design, but also identifying structural restrictions such as strict-feedback form or transformability to it (Ong et al., 2024).

Activated Backstepping CBFs (ABCs) combine these two strands. Instead of always subtracting the quadratic penalty $h_3$5, they define a switching signal

$h_3$6

and the activated barrier

$h_3$7

When $h_3$8, the real dynamics are evolving at least as safely as the virtual safe dynamics, so $h_3$9 and the barrier reduces to the original constraint. When $h_{i+1}=c_i h_i + L_f h_i$0, an activation penalty is turned on. This yields less conservative safe sets in the state space than standard CBF backstepping and eliminates the $h_{i+1}=c_i h_i + L_f h_i$1-tuning that appears in ReCBFs (Gacsi et al., 28 Aug 2025).

The comparison between these constructions has sharpened several common misconceptions. Backstepping CBFs are not identical to HOCBFs, because they synthesize a standard CBF on the full state rather than a hierarchy of auxiliary inequality constraints. Nor are they uniformly less conservative than rectified methods; the current literature instead shows that standard quadratic backstepping can be conservative, and that activation or rectification can recover safe sets closer to the original constraint in the examples studied, including double integrators, inverted pendula, and aircraft pitch constraints (Ong et al., 2024).

4. Extensions beyond nominal finite-dimensional ODEs

One extension treats disturbances through smooth Robust CBF backstepping. For systems

$h_{i+1}=c_i h_i + L_f h_i$2

the disturbance term $h_{i+1}=c_i h_i + L_f h_i$3 in the Robust CBF condition is non-smooth under repeated differentiation. The smooth robust construction replaces it with

$h_{i+1}=c_i h_i + L_f h_i$4

and builds a backstepping chain

$h_{i+1}=c_i h_i + L_f h_i$5

The final $h_{i+1}=c_i h_i + L_f h_i$6 is then enforced through a single affine QP constraint. In the unicycle example with an unknown moving obstacle, this robust backstepping filter prevents collision using only an upper bound on obstacle speed, whereas standard non-robust backstepping is reported to collide around $h_{i+1}=c_i h_i + L_f h_i$7 s in the simulation setup given in the paper (Kim et al., 2024).

A second extension concerns infinite-dimensional systems. In the Stefan problem with actuator dynamics, the main energy-deficit barrier functional

$h_{i+1}=c_i h_i + L_f h_i$8

has relative degree two with respect to the actuator input $h_{i+1}=c_i h_i + L_f h_i$9, while $SO(3)$0 has relative degree one. The paper constructs an auxiliary CBF

$SO(3)$1

and chooses the non-overshooting control

$SO(3)$2

so that $SO(3)$3. This finite-dimensional backstepping of barrier variables is combined with a PDE backstepping transformation on the temperature field and moving interface, producing a QP safety filter that guarantees the liquid phase does not freeze while remaining as close as possible to an operator input (Koga et al., 2021).

A third extension addresses geometric mechanics on manifolds. For a simple mechanical system on a Riemannian manifold $SO(3)$4, with a configuration safety function $SO(3)$5 and a safe configuration-level velocity field $SO(3)$6, the lifted backstepping CBF is

$SO(3)$7

where $SO(3)$8 is the actuated distribution and the norm is induced by the kinetic-energy metric. The theorem requires $SO(3)$9, so that unactuated directions do not affect the configuration constraint. This geometric construction avoids explicit computations on higher-order tangent bundles and is demonstrated on an underactuated satellite on $h_0$0 (Sa et al., 23 Oct 2025).

Constant-sum high-order barriers form another specialized variant. For parallel boundaries represented by $h_0$1 and $h_0$2, the backstepping chain preserves the constant-sum structure: $h_0$3 This is used to enforce “stay between parallel boundaries” constraints without collapsing them into a single barrier whose gradient vanishes at the midline, a degeneracy identified as inherent in the single-CBF encoding for symmetric strips (Kim et al., 21 May 2025).

5. Representative application domains

Backstepping CBFs have been applied across a wide range of safety-critical systems. In multicopters, the dynamics were reformulated to handle mixed-relative-degree and non-strict-feedback-form structure by augmenting total thrust and introducing a force variable $h_0$4. A single QP with affine inequality constraints then enforces angular-velocity safety, thrust-direction safety, velocity safety, and position safety simultaneously, without a cascade control design (Kim et al., 2023).

In automated vehicles, activated backstepping was implemented on a kinematic bicycle model with output $h_0$5 and obstacle-avoidance constraint

$h_0$6

A single-integrator safety filter first constructs the virtual safe controller $h_0$7, after which the activated backstepping CBF is enforced through a QP around a lane-keeping controller. The reported behavior is that the barrier remains inactive when the vehicle is far from the obstacle, activates when the approach becomes unsafe, and deactivates again after the obstacle is passed (Gacsi et al., 28 Aug 2025).

In multi-robot cooperative herding, evaders are influenced only indirectly by herder motion through inverse-power repulsive interactions, which makes the overall system underactuated. The method introduces an intermediate evader-velocity state $h_0$8, designs separate barrier functions for goal reaching and collision avoidance, and then constructs backstepping CBFs such as

$h_0$9

The resulting controller combines a nominal Sontag-type law for herding completion with CBF-QP safety filtering, and centralized as well as decentralized implementations are reported (Li et al., 14 Jul 2025).

In manufacturing-oriented PDE control, the Stefan-model safety filter is interpreted as analogous to an automotive override: it allows operator heat-and-cool commands through when safe, but saturates them between backstepping-designed limits $\psi$0 and $\psi$1 when freezing or overshoot would occur. Simulations for powder bed metal additive manufacturing are reported to show that the melt pool depth converges to its setpoint without overshoot and that the liquid never freezes (Koga et al., 2021).

In robust navigation with unknown moving obstacles, the smooth robust backstepping chain is used on an augmented unicycle model so that steering and acceleration both enter a homogeneous relative-degree-two barrier construction. The robust safety filter first retreats to increase distance, then waits for the obstacle to pass, and finally resumes the nominal trajectory (Kim et al., 2024).

6. Limitations, misconceptions, and open directions

A recurring limitation is structural. Classical safe backstepping assumes strict-feedback form or a representation that can be recursively decomposed into virtual-control layers. Later papers repeatedly note that this can be restrictive for general nonlinear systems, mixed-input relative degree, or systems whose natural coordinates do not expose a suitable cascade. Reformulations such as force augmentation in multicopters, velocity augmentation in unicycles, and distribution-based geometric splitting on manifolds are therefore not incidental technicalities; they are often prerequisites for applying the method (Kim et al., 2023).

Another recurrent issue is conservatism. Standard backstepping CBFs subtract a quadratic tracking-error penalty everywhere, so the safe set may be substantially smaller than the original top-level constraint. Rectified and activated variants were introduced precisely to localize this shrinkage to states where the actual dynamics are evolving less safely than the virtual safe dynamics. This suggests that the central design question is not whether backstepping is valid—it is—but how much safe-set contraction is acceptable for a given application (Gacsi et al., 28 Aug 2025).

It is also inaccurate to treat all high-order safety methods as interchangeable. HOCBFs, ReCBFs, backstepping CBFs, constant-sum chains, and auxiliary-barrier PDE constructions solve related problems with materially different safe-set definitions, regularity properties, and feasibility mechanisms. The literature identifies weak-relative-degree singularities, gradient-vanishing pathologies for single barriers between parallel boundaries, and the non-smoothness of robust norms under repeated differentiation as specific failure modes that motivate one construction over another (Ong et al., 2024).

Current research directions follow these fault lines. ReCBFs explicitly suggest extending rectification ideas to backstepping designs; the geometric theory suggests broader manifold-based synthesis for robotics and aerospace; the PDE work points toward more general ODE–PDE safety filters; and the constant-sum framework leaves open the treatment of non-parallel boundaries, input constraints, and more general multi-boundary configurations (Sa et al., 23 Oct 2025). A plausible implication is that “backstepping CBFs” will remain an umbrella term for a constructive design philosophy rather than a single canonical formula: start from a lower-order or lower-dimensional safety description, recursively lift it through the system architecture, and enforce the resulting first-order barrier inequality with a minimally invasive safety filter.