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SC3BF: Shifted Collision Cone Control Barrier Function for Dynamic Obstacle Avoidance
Amin Kashiri, Yasin Yazıcıoğlu
TL;DR
Velocity-space collision-cone barriers reject even slow approaches, motivating a less conservative safety condition. SC3BF adds a state-dependent allowance and enforces it with an ordinary quadratic program. In simulations with up to 100 moving obstacles, it outperformed three baselines on goal-reaching and nominal-input modification, while its safe set is forward invariant under bounded inputs.
Problem
The collision cone rejects every relative velocity aimed into an obstacle, however slow, while DPCBF's guarantees rely on minimum forward speed and clearance assumptions its controller does not enforce.
Source: §I · p. 1, C. Related Work: Velocity-Space CBFs · p. 3Method
Results
In simulations with up to 100 moving obstacles, SC3BF reached the goal more often than C3BF, DPCBF, and VOCBF and modified the nominal input less than half as much as any of them.
Source: §V · p. 5Takeaways & Limitations
The safe set is forward invariant under bounded inputs, including at contact, and a nonzero safety-preserving allowance exists for every choice of bicycle and obstacle parameters.
Source: §V · pp. 4–5
Abstract
from arXiv · showhide
The collision cone used by velocity-space control barrier functions is conservative: it rejects every relative velocity aimed into an obstacle, however slow. We propose the \emph{shifted collision-cone CBF} (SC3BF), which adds a state-dependent \emph{allowance} to the cone condition, so the robot may approach the obstacle at a rate that grows with distance and with its own speed. SC3BF is enforced by an ordinary quadratic program, and its safe set is forward invariant under bounded inputs without a minimum forward speed or a clearance margin. We prove that a nonzero allowance preserving safety always exists, and derive one in closed form. Against three velocity-space baselines on a kinematic bicycle among up to $100$ moving obstacles, SC3BF reaches the goal more often and modifies the nominal input less than half as much.
I. INTRODUCTION
The introduction presents SC3BF as a less conservative velocity-space barrier and reports safety guarantees and stronger simulation performance than three baselines. It also motivates CBFs as safety filters enforced through a small quadratic program.
A. Control Barrier Functions and the CBF-QP
- CBFs represent safety as a nonnegative state function and constrain its rate of decrease; for input-affine dynamics, the constraint is linear in the input and can be enforced by a small QP.A common safety-filter setup modifies a nominal controller's input as little as possible.Source: §I · p. 1
I. INTRODUCTION
- Distance-based barriers may not impose braking, braking-distance barriers exclude some safe steering maneuvers, and high-order CBFs certify a subset of the safe set.High-order CBFs also require auxiliary class-K functions that are difficult to tune.Source: §I · p. 1
- SC3BF shifts the collision cone with a state-dependent allowance, permitting approach rates that grow with distance and robot speed without requiring a minimum forward speed or clearance margin.The introduction contrasts this with C3BF [5], which rejects every velocity aimed into the obstacle, and DPCBF, whose guarantees rely on assumptions its controller does not enforce.Source: §I · p. 1
- SC3BF reached the goal more often and modified the nominal input less than half as much as C3BF, DPCBF, and VOCBF in simulations with up to 100 moving obstacles.The paper also proves forward invariance under bounded inputs, including at contact, and establishes that a nonzero safety-preserving allowance exists for every choice of bicycle and obstacle parameters.Source: §I · p. 1
B. Bicycle Dynamics, Obstacle Model, and Line-of-Sight Frame
The paper models the robot as a kinematic bicycle with bounded acceleration and steering input, and models each obstacle as a moving disk. Relative motion and the line-of-sight frame provide quantities used to describe approach and transverse motion.
B. Bicycle Dynamics, Obstacle Model, and Line-of-Sight Frame
- The kinematic bicycle state is position, heading, and forward speed, with acceleration and front-wheel steering as inputs; under the small-slip approximation, the input is acceleration and slip angle.The admissible bounds are |a| ≤ amax and |β| ≤ βmax.Source: B. Bicycle Dynamics, Obstacle Model,… · p. 2
- Each dynamic obstacle is a disk, and the combined safety radius sums robot and obstacle radii; relative position and velocity define distance and a worst-case approach-speed bound.The obstacle's position is augmented into the state, and the robot's forward speed is assumed to lie in [0, vmax] for the feasibility result.Source: B. Bicycle Dynamics, Obstacle Model,… · p. 2
- The line-of-sight frame aligns its first axis with relative position, so relative velocity's parallel component is the closing rate and its transverse component is perpendicular to that direction.The heading in this frame is measured relative to the line of sight.Source: B. Bicycle Dynamics, Obstacle Model,… · p. 2
C. Related Work: Velocity-Space CBFs
The related-work section compares velocity-space CBFs: collision cones are conservative, parabolic regions reduce that conservatism under assumptions, and disjunctive cone-edge constraints use a mixed-integer formulation. It also notes that these methods do not address the barrier's contact singularity.
C. Related Work: Velocity-Space CBFs
- C3BF keeps relative velocity outside the collision cone, a condition with relative degree one for the kinematic bicycle, but its cone ignores relative-velocity magnitude.The cone comprises directions that would collide under constant-velocity motion.Source: C. Related Work: Velocity-Space CBFs · p. 2
- DPCBF replaces the cone with a parabolic region depending on distance and relative-velocity magnitude, but its guarantees require minimum forward speed and a clearance margin.The controller assumes rather than enforces the clearance margin, and the speed assumption excludes zero forward speed.Source: C. Related Work: Velocity-Space CBFs · pp. 2–3
- VOCBF enforces at least one of the two cone-edge half-space barriers through a disjunction; the compared formulation is mixed-integer.The related-work discussion says its mixed-integer formulation scales poorly with obstacle count.Source: C. Related Work: Velocity-Space CBFs · p. 3
- These velocity-space methods share a contact limitation: d = ρ² − r² has a derivative singular at contact; verifies condition (1) only for d > 0, assumes no contact, and bounds the term by (A2).Source: C. Related Work: Velocity-Space CBFs · p. 3
III. PROPOSED METHOD
SC3BF shifts the collision-cone condition using an allowance that depends on distance and robot speed. Its controller enforces the barrier away from contact and uses contact constraints where the barrier derivative is singular.
A. Proposed Barrier Function
- The allowance lets the robot approach at rates up to c, which vanishes at contact and rest and grows with distance and robot speed.The cone condition applies to the excess closing rate.Source: A. Proposed Barrier Function · p. 3
- The barrier is the maximum of two edge functions; κ = 0 recovers the C3BF safe set, while positive κ enlarges it.The safe set also requires ρ ≥ r.Source: A. Proposed Barrier Function · p. 3
- In the relative-velocity plane, c shifts the cone apex along the closing direction, and the gray region marks the conservatism removed.Source: Figure 1 · p. 3
B. Proposed Controller
- Away from contact, the controller differentiates hσ along the augmented dynamics; the resulting expression includes terms from sideways center-of-mass motion.Source: B. Proposed Controller · p. 3
- At contact, where the barrier derivative is singular, the controller instead enforces nondecreasing distance and nonincreasing closing rate; the active edge is selected by the sign of Δv⊥, yielding an ordinary QP.Source: B. Proposed Controller · p. 3
C. Main Results
The paper establishes that an admissible allowance pair exists and that the resulting controller preserves the safe set. It also gives a closed-form threshold and a procedure for certifying larger gains on a specified domain.
C. Main Results
- An admissible pair is feasible at every state of the safe set for every extended class-K function, and a pair with κ > 0 exists for all stated positive safety radii and bounded system parameters.Source: C. Main Results · pp. 3–4
- At contact, the controller uses contact constraints that are feasible with u = 0; together with the invariance result, feasibility is maintained along the trajectory.Source: C. Main Results · p. 4
- If the applied input satisfies the controller constraints, the safe set is forward invariant and the robot maintains ρ ≥ r for all time.Source: C. Main Results · p. 4
D. Choosing the Allowance
- At p = 1, every 0 < κ ≤ κ⋆ is admissible, with κ⋆ given in closed form and no numerical search required.The threshold is conservative because the drift and input contributions are bounded separately at their respective worst cases.Source: D. Choosing the Allowance · p. 4
- A numerical procedure can certify a larger κ across every state in a specified domain; within the certified range, larger κ reduces conservatism but permits faster approaches.The procedure certifies one (κ, p) pair at a time, and p sets how quickly the allowance grows with distance.Source: D. Choosing the Allowance · p. 4
IV. EVALUATION
Across the tested obstacle counts, SC3BF reaches the goal more often than every baseline and uses less than half their intervention. The conclusion states that its safe set is forward invariant under bounded inputs, including at contact, without a clearance margin or minimum forward speed.
A. Setup and Metrics
- Each method faced the same 100 random instances at 20, 50, or 100 moving obstacles, using a shared goal-tracking nominal controller.Success meant reaching within 0.3 m of the goal; infeasibility, collision, or hitting the step limit counted as failure, and intervention was averaged over successful runs.Source: A. Setup and Metrics · p. 4
B. Results
- SC3BF success was 92% vs DPCBF’s 84%, 67% vs 48%, and 27% vs 17% across 20, 50, and 100 obstacles; intervention was under half of every baseline.Table I reports success rate and intervention over 100 random instances per obstacle count, with intervention averaged over successful runs.Source: Table I · p. 4, B. Results · p. 4
V. CONCLUSION
- The conclusion states that SC3BF’s safe set is forward invariant under bounded inputs, including at contact, without a clearance margin or minimum forward speed.The paper also reports that a nonzero safety-preserving allowance exists for any bicycle and obstacle parameters.Source: §V · pp. 4–5
APPENDIX I RATES OF THE RELATIVE MOTION AND OF AN EDGE
The appendix derives relative-motion and collision-edge rates by expressing motion in the line-of-sight frame, then differentiating the distance and edge quantities used elsewhere in the paper.
APPENDIX I RATES OF THE RELATIVE MOTION AND OF AN EDGE
- The appendix derives the rates of relative distance and velocity components, the collision-edge rate, and the rate of the velocity component along the heading.It uses the bicycle dynamics and constant obstacle velocity to express relative motion in the line-of-sight frame.Source: Appendix I · p. 5
- For ρ > r, differentiating d² = ρ² − r² gives the distance rate ˙d = ρ ˙ρ/d.This rate is used for the non-contact case.Source: Appendix I · p. 5
- For ρ > r, differentiating the edge expression with ˙v = a yields its rate, after substituting the relative-motion rates and collecting input terms.The resulting expression separates terms independent of input from coefficients of acceleration and steering.Source: Appendix I · p. 5
APPENDIX II THE CONTACT CONSTRAINTS
The appendices handle contact with separate constraints and establish a sufficient condition for a safety-preserving allowance by comparing bounds on barrier drift and control authority.
APPENDIX II THE CONTACT CONSTRAINTS
- At contact, the edge-rate formula is singular, so separate affine constraints prevent entry and preserve C; zero input satisfies both.The constraints are imposed for 0 < ρ ≤ r because the proof considers an interval with ρ < r.Source: APPENDIX II THE CONTACT CONSTRAINTS · pp. 5–6
APPENDIX III PROOF OF PROPOSITION 3.1
- For states in Cσ with ρ > r, it suffices to find an input making ˙hσ nonnegative, since −α(hσ) ≤ 0 there.The proof maximizes the affine rate over bounded inputs and treats the case ∆v∥ ≤ 0 separately from ∆v∥ > 0.Source: APPENDIX III PROOF OF PROPOSITION 3.1 · p. 6
A. Bounds on the Drift and the Authority
- Drift and control authority are bounded proportionally to ρv², yielding a parameter-only sufficient condition that includes v = 0.For p > 1, the distance range must be bounded to bound d^(p−1).Source: A. Bounds on the Drift and the Authority · pp. 6–7
B. A Closed-Form Gain
For p = 1, the proof bounds terms that depend on κ to derive an explicit positive gain limit; gains up to that limit satisfy the stated condition.
B. A Closed-Form Gain
- At p = 1, linearly bounding κ-dependent terms yields an explicit κ* > 0; every 0 < κ ≤ κ* satisfies the condition.The proof uses D̄ = 1 and M₁ = 2 for this case.Source: B. A Closed-Form Gain · p. 7
C. Completing the Proof
The closed-form gain makes the barrier constraint feasible away from contact, and the appendix proves that inputs satisfying the controller constraints preserve the safe set, including at contact.
APPENDIX IV PROOF OF THEOREM 3.2
- The proof establishes that every absolutely continuous trajectory satisfying the controller constraints almost everywhere remains in C, with ρ(t) ≥ r and h(t) ≥ 0.The closed-form gain ensures feasibility away from contact; contact constraints handle the boundary.Source: APPENDIX IV PROOF OF THEOREM 3.2 · pp. 7–8, C. Completing the Proof · p. 7
- Because h is the maximum of two edge functions, a lemma supplies its absolute continuity and almost-everywhere derivative even where the edges tie.At a tie, the derivative of the maximum equals the derivative of either active edge.Source: APPENDIX IV PROOF OF THEOREM 3.2 · p. 7
- At contact, the constraints imply a nondecreasing distance, ruling out any transition to ρ < r.The proof uses absolute continuity of ρ on the interval where the contact constraints apply.Source: APPENDIX IV PROOF OF THEOREM 3.2 · p. 7
- Away from contact the barrier constraint controls h, while at contact −r∆v∥ equals h and is nondecreasing, so h cannot cross below zero.When h is negative away from contact, the extended class-K condition makes its derivative positive almost everywhere.Source: APPENDIX IV PROOF OF THEOREM 3.2 · pp. 7–8
APPENDIX V CERTIFICATION OF THE GAIN
Appendix V certifies whether a selected gain and exponent satisfy the feasibility condition throughout a specified domain. Its bounds can be conservative, and for p > 1 certification is limited to a bounded distance range.
APPENDIX V CERTIFICATION OF THE GAIN
- The closed-form gain κ⋆ is conservative because it separately bounds drift and authority using worst cases that may not occur at the same state.For p > 1, verification is performed over a specified range ρ ≤ ρmax.Source: Appendix V · p. 8
- At p = 1 the constants are D̄ = 1 and M1 = 2; for p > 1, the bounds use a distance range capped at ρmax.The p > 1 constants bound terms involving d^(p−1) over that range.Source: Appendix V · p. 8
- For low speeds, bisection finds the largest verified cutoff vc; the pair is not verified if no vc > 0 satisfies the condition.A verified cutoff establishes the condition for all speeds v ≤ vc.Source: Appendix V · p. 8
- At remaining speeds, interval arithmetic bounds each box; nonnegative bounds certify boxes, which are subdivided until the domain is covered or verification fails.Failure occurs if a box reaches minimum width with a negative bound or the list exceeds its size limit.Source: Appendix V · p. 8