Model-Free Safety-Critical Control for Robotic Systems

I. INTRODUCTION

• High-fidelity robot dynamics are often complicated and high-dimensional, making safety-critical controller synthesis and implementation nontrivial, including CBF design and safe-input calculation.Prior work has explored data-driven barrier functions [7], robust CBFs, [9], and reduced-order models –.Source: §I · p. 1
• The method uses CBFs to design a safe velocity from reduced-order kinematics and tracks it, avoiding reliance on the robot’s full-order dynamics for safety,.The tracking controller may still require system-specific knowledge of the full model, but safety enforcement after stable velocity tracking is established does not require the high-fidelity model.Source: §I · p. 1
• The paper formalizes and generalizes the approach with a theoretical safety proof and demonstrations on a Segway in simulation and a Drone and Quadruped in hardware.The authors present the applications as evidence of the method’s domain agnosticism.Source: §I · p. 1

II. PRELIMINARIES

• The preliminaries motivate the framework through stable tracking of a safe velocity and introduce CLFs and CBFs as tools for stability and safety guarantees.The text describes these functions alongside stable and safe trajectories in Fig. 2.Source: §II · p. 1
• The setup considers control-affine systems with Lipschitz dynamics and controllers, assumes solutions exist for all time, and treats the origin as an equilibrium in an open connected neighborhood.It also defines class-K functions as strictly increasing functions that vanish at zero.Source: §II · pp. 1–2

A. Stability and Control Lyapunov Functions

• Exponential stability requires trajectories from sufficiently small initial conditions to decay at an exponential rate over time.The definition bounds the state norm by M e^-βt times the initial-state norm for ∥x0∥ ≤ a.Source: A. Stability and Control Lyapunov… · p. 2
• A continuously differentiable CLF provides an efficient route to exponential stability: any locally Lipschitz controller satisfying the stated condition renders the origin exponentially stable.The passage identifies this controller condition as the basis for synthesizing an input that enforces exponential stability.Source: A. Stability and Control Lyapunov… · p. 2

B. Safety and Control Barrier Functions

• A system is safe with respect to S when S is forward invariant: starting in S ensures the state remains in S for all t ≥ 0.The safe set is application-driven and may represent robot positions that avoid obstacle collisions; here it is defined as the 0-superlevel set of a continuously differentiable function h.Source: B. Safety and Control Barrier Functions · p. 2
• A control barrier function provides a tool for synthesizing provably safe controllers, analogous to how control Lyapunov functions support stability.The figure contrasts a CBF and safe trajectory with a CLF and stable trajectory; h may take any real value, unlike nonnegative V.Source: B. Safety and Control Barrier Functions · p. 2, Figure 2 · p. 2
• The figure distinguishes ISS from ISSf: ISS trajectories converge to a neighborhood of x = 0, while ISSf makes a superset Sd of S forward invariant.It depicts these conditions alongside the stability and safety conditions.Source: Figure 2 · p. 2
• Theorem 2 establishes safety-critical controller synthesis through condition (8), and a desired controller can be minimally modified by solving a quadratic program.The controller is locally Lipschitz continuous; its Lipschitz continuity is discussed in.Source: B. Safety and Control Barrier Functions · p. 2

C. Effect of Disturbances

• A bounded disturbance d added to the input u changes the system to ẋ = f(x) + g(x)(u + d).The section motivates disturbance-aware guarantees because unknown disturbances may compromise robotic-system stability or safety.Source: C. Effect of Disturbances · p. 2
• Exponential ISS bounds trajectories by a decaying initial-state term plus a disturbance-dependent term, so solutions converge to a neighborhood of the origin;, show it follows by strengthening Theorem 1’s condition (4).The section specifies that the bound uses a class-K function of the disturbance magnitude.Source: C. Effect of Disturbances · p. 3
• ISSf requires trajectories starting in a disturbance-dependent neighborhood S_d of the safe set S to remain in S_d; guarantees it by replacing Theorem 2’s condition (8).The neighborhood is defined as a 0-superlevel set using a class-K function.Source: C. Effect of Disturbances · p. 3

III. MODEL-FREE SAFETY-CRITICAL CONTROL

• Safety requires q(t) to remain in S for all t≥0, where S is the 0-superlevel set of a continuously differentiable function h that depends only on configuration.The stated control problem is to achieve this safety for specified initial conditions.Source: §III · p. 3

A. Control Method

• The method synthesizes and tracks a safe velocity, reducing safety-critical control complexity and using a kinematic condition rather than the full robot dynamics.The approach is described as enabling model-free safety-critical control.Source: A. Control Method · p. 3
• The synthesized safe velocity depends on configuration and satisfies a kinematic safety condition parameterized by α>0.That condition does not depend on the full robotic dynamics.Source: A. Control Method · p. 3
• A velocity-tracking controller tracks the safe velocity, with exponential error convergence assumed; the exposition assumes the safe velocity has a second derivative and later addresses disturbances.The smoothness assumption is later relaxed in Remark 4.Source: A. Control Method · p. 3
• In the double-integrator example, a proportional desired velocity toward the goal is modified minimally through a CBF-based quadratic program to avoid an obstacle.The safe velocity can be tracked with u=−KD( ˙q−˙qs).Source: A. Control Method · p. 3
• With α=0.1 or 0.2 s^-1, the double integrator avoided two obstacles, while α=0.5 or 1 s^-1 violated safety.The simulations used KP=0.2 s^-1 and KD=1 s^-1; the figure indicates safety when α is small enough.Source: A. Control Method · pp. 3–4, Figure 3 · p. 4
• For multiple obstacles, the method considers the closest obstacle at each time, making the CBF nonsmooth and the safe velocity only piecewise differentiable.The nonsmooth CBF has been analyzed in, and the constructions are restricted to differentiable segments.Source: A. Control Method · p. 4

B. Main Result

• When λ > α, initial membership in SV guarantees q(t) remains in S for all t ≥ 0 under the specified safe velocity and velocity-tracking controller.SV is defined by hV(q, ˙e) = −V(q, ˙e) + αe h(q) ≥ 0.Source: B. Main Result · p. 4
• Choosing α small enough makes λ > α achievable for a given tracking controller, but smaller α can make the system more conservative by keeping it farther from the safe-set boundary.The paper suggests gradually increasing α from 0; the theorem guarantees the existence of a suitable α.Source: B. Main Result · p. 4
• The safe-velocity condition can use reduced-order dynamics ˙q = A(q)µs, including a unicycle model, provided the tracking controller satisfies the required error-stability property.The safe input satisfies ∇h(q)A(q)µs ≥ −αh(q), with tracking error ˙e = ˙q − A(q)µs.Source: B. Main Result · p. 4
• The initial-state requirement h(q0) ≥ V(q0, ˙e0)/αe is stricter than merely requiring q0 ∈ S, but becomes less conservative with smaller tracking error or faster tracking.Faster tracking corresponds to a larger λ − α and therefore a larger αe.Source: B. Main Result · p. 4
• Theorem 3 extends to non-differentiable tracking errors when their norm decays exponentially at rate λ, yielding a comparison bound that keeps h(q(t)) nonnegative.The extension assumes ∥˙e(t)∥ ≤ M∥˙e0∥e^−λt.Source: B. Main Result · p. 5

C. Effect of Disturbances

• Bounded disturbance d models imperfect tracking, delays, or modeling errors, and ISSf guarantees invariance of the enlarged set Sd defined by hd(q) = h(q) + γ(∥d∥∞).The dynamic extension SVd contains the safe-set states paired with tracking error velocities that satisfy hVd(q, ˙e) ≥ 0.Source: C. Effect of Disturbances · p. 5
• ISS tracking requires ˙V ≤ −λV + ι(∥d∥∞); if tracking error does not decay, the bound reduces to ∥˙e(t)∥ ≤ ∥˙e∥∞, recovering traditional ISSf guarantees in.Exponential ISS is the strongest assumption considered, while poor tracking makes the disturbance-dependent term dominate the tracking-error bound.Source: C. Effect of Disturbances · p. 5
• When λ > α, initial states in the dynamic safe set imply q(t) remains in the disturbance-expanded safe set for all t ≥ 0, with γ(∥d∥∞) = ι(∥d∥∞)/α.Thus, input-to-state stable tracking of a safe velocity yields input-to-state safety for the full system.Source: C. Effect of Disturbances · p. 5

D. Velocity Tracking Controllers

• The simplest option is a model-free D controller; when model-dependent terms are known, gravity compensation is also available for fully actuated systems with n = m and invertible B.The section also introduces a heavily model-dependent extension, with controller-design parameters α and KD or K.Source: D. Velocity Tracking Controllers · p. 5
• A heavily model-dependent extension may improve tracking but requires D(q) and C(q, ˙q), whose expressions may be complicated and expensive to compute.Source: D. Velocity Tracking Controllers · p. 5
• The parameter λ characterizes tracking speed as the smallest gain divided by the largest inertia.The section associates the controllers with a Lyapunov-function candidate.Source: D. Velocity Tracking Controllers · p. 5
• Controllers (29), (30), and (31) satisfy distinct ISS or stability conditions: (29) is ISS, (30) is ISS for moving references and stable for stationary references, and (31) is stable.For (29), the disturbance is d = −D(q)¨qs − C(q, ˙q) ˙qs − G(q); for (30), it is d = −D(q)¨qs − C(q, ˙q) ˙qs when ˙qs ≠ 0.Source: D. Velocity Tracking Controllers · p. 5

• The applications include a Segway in high-fidelity simulation and a Drone and Quadruped in hardware experiments.Source: §IV · p. 6

A. Numerical Simulation of Segway

• In the planar simulation, the Segway tracks a desired 1 m/s forward velocity and stops safely in front of a wall at pmax = 2 m.The simulation starts at p0 = 0 with α = 0.5 s−1.Source: A. Numerical Simulation of Segway · p. 6
• The spatial Segway reaches its goal safely while tracking a unicycle-model safe velocity derived using a heading-aware CBF and incorporated into a quadratic program.The tracked safe velocity is μs = (vs, ωs), with the reported simulation using Kv = 0.16 s−1, Kω = 0.8 s−1, α = 0.2 s−1, δ = 0.5 m, and R = 0.25 m.Source: A. Numerical Simulation of Segway · p. 7
• The spatial tracking controller is model-free, while its gains are tuned to stabilize velocity tracking using the full dynamics, based on linearization and LQR.The planar controller is likewise described as not relying on the full dynamics; its gains also stabilize the Segway upright.Source: A. Numerical Simulation of Segway · pp. 6–7

B. Hardware Experiments on Drone and Quadruped

• The experiments used known obstacle locations and sensory information only to determine robot position, with safe velocities based on single-integrator and unicycle models.The obstacle-avoidance task was executed on two fundamentally different hardware platforms: a Drone and a Quadruped.Source: B. Hardware Experiments on Drone and… · p. 7
• Both the Drone and Quadruped completed the obstacle-avoidance task safely while tracking single-integrator safe velocities with platform-specific controllers.For the Drone, α = 0.2 s^-1 was selected small enough for available tracking performance and was not tuned for optimal performance.Source: B. Hardware Experiments on Drone and… · pp. 7–8
• The Quadruped’s safe single-integrator experiment used forward- and side-stepping, indicating the model-free approach’s application-agnostic nature despite its fundamentally different dynamics from the Drone.The Quadruped’s yaw rate was set to zero for this experiment.Source: B. Hardware Experiments on Drone and… · p. 8
• With a unicycle-based safe velocity, the Quadruped traversed the obstacle course safely by walking forward and turning rather than forward- and side-stepping.The safe forward velocity and yaw rate were tracked by the same ID-QP walking controller; safety was provably guaranteed in a model-free fashion.Source: B. Hardware Experiments on Drone and… · p. 8

V. CONCLUSIONS

• Following, the method synthesizes a safe velocity with control barrier functions and tracks it, guaranteeing safety when tracking is faster than safe motion approaches the safe-set boundary.The method does not require the robot’s full dynamical model to achieve safety.Source: §V · p. 8
• The model-free approach is application-agnostic, with obstacle avoidance demonstrated for a Segway in simulation and a Drone and Quadruped in hardware.Source: §V · p. 8
• The method still relies on kinematic models such as single-integrator or unicycle models; future work will explore richer reduced-order models and relax the velocity-tracking performance assumption.Source: §V · p. 8