Source-linked AI summary
Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function
Jun Zeng, Bike Zhang, Koushil Sreenath
TL;DR
Robotic MPC must optimize near state and input limits while keeping systems inside safe invariant sets. The paper proposes MPC-CBF, combining receding-horizon control with discrete-time CBF constraints, and validates it through obstacle avoidance and racing scenarios. The method is reported to outperform MPC-DC and DCLF-DCBF and to support safe overtaking in competitive racing.
Problem
MPC handles operational constraints, but safety-critical robotic control still requires guaranteeing that the system remains within an invariant set.
Method
MPC-CBF unifies model predictive control with discrete-time control barrier functions in one safety-critical optimization design.
Results
MPC-CBF is reported to outperform MPC-DC and DCLF-DCBF, with validation on 2D obstacle avoidance and competitive racing where the ego car safely overtakes other cars.
Takeaways & Limitations
The design combines predictive performance with discrete-time CBF-based safety in set-invariance problems and extends from a double integrator to competitive car racing.
Abstract
from arXiv · showhide
The optimal performance of robotic systems is usually achieved near the limit of state and input bounds. Model predictive control (MPC) is a prevalent strategy to handle these operational constraints, however, safety still remains an open challenge for MPC as it needs to guarantee that the system stays within an invariant set. In order to obtain safe optimal performance in the context of set invariance, we present a safety-critical model predictive control strategy utilizing discrete-time control barrier functions (CBFs), which guarantees system safety and accomplishes optimal performance via model predictive control. We analyze the stability and the feasibility properties of our control design. We verify the properties of our method on a 2D double integrator model for obstacle avoidance. We also validate the algorithm numerically using a competitive car racing example, where the ego car is able to overtake other racing cars.
I. INTRODUCTION
The paper unifies model predictive control with discrete-time control barrier functions to pursue safe, predictive control for robotic tasks. It analyzes stability and feasibility, then validates the design in obstacle avoidance and competitive racing.
- C. Contribution: MPC-CBF directly unifies model predictive control with discrete-time control barrier functions for safety-critical tasks.The formulation enforces safety-critical constraints through discrete-time CBFs.
- A. Motivation: Euclidean-distance obstacle constraints may not confine robots until they are close to obstacles, while longer MPC horizons increase optimization time.This motivates CBF constraints that constrain movement during optimization.
- C. Contribution: The method analyzes stability and feasibility through reachable-set intersections with safe sets along the prediction horizon.The paper also discusses relations with MPC-DC and DCLF-DCBF.
- C. Contribution: MPC-CBF is reported to outperform both MPC-DC and DCLF-DCBF by adding prediction capability to DCLF-DCBF.The comparison is verified using a 2D double integrator for obstacle avoidance.
- C. Contribution: In competitive racing, MPC-CBF enables an ego car to race on a track while safely overtaking other cars.The example includes overtaking from both left and right directions while maintaining a specified target speed.
D. Paper Structure
The paper formulates finite-horizon optimal control from the current state, applies only the first optimized input, and repeats the problem at each subsequent step. It validates this framework through double-integrator obstacle avoidance and competitive car racing.
- A. Model Predictive Control: At each time step, the controller solves a finite-time optimal control problem using the current state estimate and system constraints.Safety criteria such as obstacle avoidance are represented through distance constraints.
- A. Model Predictive Control: The optimization includes stage and terminal costs, state and input constraints, distance-based safety constraints, and a terminal constraint.The prediction horizon is denoted by N.
- A. Model Predictive Control: Only the first element of the optimized input sequence is applied, producing a feedback control law and an open-loop optimized trajectory.The finite-time problem is then repeated after the next state estimate becomes available.
- A. Model Predictive Control: The resulting repeated optimization is the model predictive control strategy, whose applied-input trajectory is called the closed-loop trajectory.Each optimization uses the newly estimated state at the next time step.
B. Control Barrier Functions
Discrete-time control barrier functions define safe-set conditions that constrain the evolution of a barrier function, while a unified DCLF-DCBF program combines safety with stabilization. Under stated initial and parameter conditions, the safe set is invariant.
- B. Control Barrier Functions: The safe set C is defined as the superlevel set of a continuously differentiable function h.The paper refers to C as the safe set.
- B. Control Barrier Functions: A discrete-time CBF condition constrains the change in h so that h(x_k+1) ≥ (1 − γ)h(x_k).Thus, the lower bound of h decreases exponentially at rate 1 − γ.
- B. Control Barrier Functions: A discrete-time control Lyapunov function is introduced to stabilize the system alongside the safety objective.The CLF and CBF objectives are unified in one optimization program.
- B. Control Barrier Functions: The DCLF-DCBF optimization uses a nonnegative slack variable δ that permits CLF growth when CLF and CBF constraints conflict.H(x) is positive definite and l is positive.
- B. Control Barrier Functions: If h(x_0) ≥ 0 and 0 < γ ≤ 1, the safe set C is invariant along trajectories controlled by the unified program.This connects the discrete-time barrier constraint to the paper’s safety guarantee.
III. CONTROL DESIGN
MPC-CBF formulates safety-critical control as a constrained finite-time optimal control problem, combining MPC constraints with discrete-time CBF constraints and applying the first optimized input in receding-horizon fashion.
- MPC-CBF solves a constrained finite-time optimal control problem over horizon N while regulating the system toward a target state under safety requirements.
- The formulation includes initial-condition, dynamics, state/input, terminal-set, and discrete-time CBF constraints along the prediction horizon.The CBF constraints are designed to guarantee forward invariance of the safe set C.
- The reachable-set propagation and CBF level-set construction are illustrated across horizon steps for multiple choices of γ.
- The optimal solution is a sequence of inputs, whose first element is applied to the system at the current time step.
- Repeating this optimization at the next time step with the new state yields the receding-horizon MPC-CBF strategy.
- Because discrete-time CBF constraints are generally non-convex, the overall optimization is generally a nonlinear programming problem.
B. Stability
The paper analyzes feasibility through intersections between MPC reachable sets and CBF-induced safe sets, showing that γ controls a safety–feasibility tradeoff while persistent feasibility remains unresolved.
- Feasibility analysis: Feasibility requires the intersection R_k ∩ S_cbf,k to be nonempty at every horizon step.R_k satisfies dynamics, state/input, and initial-condition constraints, while S_cbf,k satisfies the CBF constraints.
- Feasibility analysis: If h is a valid control barrier function and X is control invariant, S_cbf,k and R_k are each nonempty.
- Safety–feasibility tradeoff: Smaller γ makes S_cbf,k smaller and increases safety through slower CBF decay, but may render the intersection with R_k infeasible.
- Safety–feasibility tradeoff: Larger γ expands S_cbf,k and makes optimization more likely to be feasible, but CBF constraints may become inactive during optimization.
- Comparison with MPC-DC: MPC-CBF can confine R_k even when the robot is far from obstacles, whereas distance constraints may be inactive while R_k remains within C.
- Open limitations: For fixed γ, the formulation generally provides only pointwise feasibility; automatically choosing γ and ensuring persistent feasibility remain open problems.
D. Relation with DCLF-DCBF
MPC-CBF transfers CLF behavior into the cost while retaining discrete-time CBF constraints, yielding a smaller safe set than MPC-DC and earlier obstacle avoidance under suitable parameters.
- Relation with DCLF-DCBF: With N = 1, MPC-CBF transfers CLF constraints into the terminal cost, eliminating the slack variable needed to guarantee feasibility.
- Relation with DCLF-DCBF: MPC-CBF with N = 1 operates similarly to DCLF-DCBF because both use comparable CLF and CBF components.
- Relation with MPC-DC: MPC-CBF constraints act at the next predicted horizon step, whereas MPC-DC distance constraints act at the current horizon step.
- Relation with MPC-DC: When moving toward obstacles, MPC-CBF can confine the reachable set before it intersects unsafe regions, unlike inactive distance constraints in MPC-DC.
- Relation with MPC-DC: R_k ∩ S_cbf,k is a proper subset of R_k ∩ C, so MPC-CBF uses a smaller safe set and is safer in the context of smaller set invariance.
- Relation with MPC-DC: Smaller γ with a shorter horizon can reduce obstacle-avoidance optimization complexity while achieving performance similar to MPC-DC.
IV. EXAMPLES
The examples evaluate MPC-CBF on a discrete-time 2D double integrator, comparing it with DCLF-DCBF and demonstrating its obstacle-avoidance behavior.
- 2D Double Integrator: The authors validate MPC-CBF numerically on a 2D double integrator for obstacle avoidance and use IPOPT for the optimization problems.
- 2D Double Integrator: The double-integrator experiment uses sampling time Δt = 0.2s and state and input constraints X and U.
- Comparison with DCLF-DCBF: The DCLF-DCBF comparison uses matched input penalties, identical discrete-time CBF constraints, and a control-Lyapunov terminal cost.
- Comparison with DCLF-DCBF: With γ = 0.4, MPC-CBF at N = 1 behaves similarly to DCLF-DCBF, while N = 8 avoids earlier and uniquely reaches the goal within the limited simulation time.
2) Comparison with MPC-DC:
MPC-CBF is compared with MPC-DC using obstacle-avoidance trajectories and benchmarks, then applied to vehicle racing with dynamic competing cars.
- Comparison with MPC-DC: The MPC-DC comparison replaces the discrete-time CBF constraint with a Euclidean-norm distance constraint using the same obstacle function.
- Comparison with MPC-DC: As γ decreases, MPC-CBF avoids earlier and creates a smaller safe set, while MPC-DC follows the trajectory closest to the obstacle; γ = 1 produces nearly identical trajectories.
- Comparison with MPC-DC: MPC-CBF with N = 5 and γ = 0.25 turns similarly to MPC-DC with N = 7, indicating comparable behavior with a shorter horizon.
- Comparison with MPC-DC: The benchmark compares prediction horizon, computation time, minimal obstacle distance, and trajectory cost integral.
- Comparison with MPC-DC: MPC-CBF uses less prediction horizon and computation time, maintains a γ-dependent safety margin, and generally has lower trajectory cost than MPC-DC in this scenario.
- Vehicle Model: The racing example uses curvilinear vehicle dynamics with longitudinal acceleration and steering inputs to model an ego car competing against other cars.
2) Control Design:
MPC-CBF combines centerline tracking and target-speed objectives with discrete-time CBF constraints for safely overtaking other racing cars. In simulation, the ego car tracks the desired speed while overtaking cars on both sides.
- Car Racing Control Design: MPC-CBF uses a stage cost that tracks the centerline at target speed vt while minimizing centerline tracking error.The target state is xr = (vt, 0, 0, 0, 0, 0), with vt = 0.6 m/s in the racing simulation.
- Car Racing Control Design: CBF constraints encode the safety criteria for overtaking each other racing car in curvilinear coordinates.Each CBF represents the safety relationship between the ego car and one other car using their longitudinal and lateral positions.
- Car Racing Control Design: The speed profile compares the ego car’s speed with the desired speed vt = 0.6m/s during one simulated lap.The dashed black line denotes the desired speed.
- Car Racing Control Design: N = 12 and 10 Hz define the MPC horizon and update rate, while the vehicle dynamics are simulated at 1000 Hz.The controller sampling time is 0.1 s, and the control design uses linearized dynamics along the centerline.
- Car Racing Results: The ego car safely overtakes racing cars with both right and left turns while attempting to catch the target speed.Opposite distance deviations of the other cars from the centerline produce overtaking motions in opposite directions.
- Car Racing Results: The paper reports that the safety-critical MPC design uses discrete-time CBF constraints in receding-horizon optimization to ensure safety.The design is also applied to competitive racing, where the ego car safely overtakes other cars.
A. Car Racing Implementation
The car-racing implementation uses a curvilinear vehicle model but replaces its highly nonlinear dynamics with data-driven, time-invariant linearized dynamics. This approximation is used in the MPC-CBF constraints for computational tractability.
- A. Car Racing Implementation: The vehicle model uses curvilinear coordinates under a Frenet reference frame and Euler discretization.The model includes track curvature and front- and rear-tire lateral forces.
- A. Car Racing Implementation: Tire-angle relations connect the front and rear tire angles to the vehicle’s states and inputs.The vehicle parameters include axle distances, tire-model coefficients, mass, and moment of inertia.
- A. Car Racing Implementation: The highly nonlinear kinematic model cannot be used directly for the system-dynamics constraints in the MPC-CBF formulation.The implementation therefore first simulates centerline tracking with a PID controller.
- A. Car Racing Implementation: The PID controller tracks the track centerline at a user-defined positive target speed before the linearized dynamics are constructed.The target speed vd is user-defined in this preliminary trajectory-generation step.
- A. Car Racing Implementation: A regression-based approach identifies time-invariant linearized dynamics from the PID-generated trajectory.The resulting approximation is used in CBF constraints and reduces computational complexity without prior knowledge of system parameters.