Safety-Critical Control under Uncertainty via Adaptive Conformal Quantile Prediction Intervals
Authors: Hao Zhou, Yanze Zhang, Yiwei Lyu, Wenhao Luo
Organizations: Department of Computer Science, University of Illinois Chicago, Chicago, IL, 60607, USA · Department of Computer Science and Engineering, Texas A&M University, College Station, TX, 77843, USA
Safety-critical control under uncertainty requires uncertainty representations that are both statistically valid (for certifiable performance) and compatible with enforceable safety constraints. However, existing methods often assume particular distributions of uncertainty for provable safety guarantees or establish symmetric and input-agnostic prediction intervals for robust safety, which can lead to misaligned or overly conservative safety constraints in control synthesis. In this paper, we introduce a novel safe control framework with adaptive uncertainty quantification that constructs calibrated and state-dependent prediction intervals to enable high-probability safety guarantees, while improving constrained control performance. The framework leverages adaptive conformal prediction (ACP) and extends it with conformal quantile regression (CQR) to capture distribution-free, asymmetric uncertainty intervals with certifiable probabilistic coverage, and integrates the resulting uncertainty sets into a probabilistic control barrier function formulation to enforce robust safety with reduced conservativeness. This yields uncertainty-aware safe control constraints that can be incorporated within a model predictive control(MPC) framework to provide provably safe behaviors with high probability. Simulation and theoretical results are provided to demonstrate the effectiveness of our approach.
Figures & tables
Fig. 1: CQR score computation under MPC. Considering the prediction horizon H=3 at time step tk−1 . Let Ftk−1−τ=F(xˉk−1∣k−1−τ,uk−1∣k−1−τ) , F^tk−1l=F^αl(x^k−1∣k−1,uk−1∣k−1;dαl) , and F^tk−1u=F^αu(x^k−1∣k−1,uk−1∣k−1;dαu) . The time-lagged CQR score is defined as Sk−1+τ∣k−1=max(F^tk−1l−Ftk−1−τ,Ftk−1−τ+F^tk−1u) for τ=0,1,…,H−1 . The resulting error terms Sk−1+τ∣k−1 are then added into the conformal score set S .
Fig. 2: Residual score computation with DT-CBF under MPC. For the current prediction horizon ( H=3 ) starting at time step tk−1 , the time-lagged CBF residual is defined as Sk−1+τ∣k−1rs=∣F(x^k−1∣k−1,uk−1∣k−1)−F(xˉk−1∣k−1−τ,uk−1∣k−1−τ)∣ for τ=1,…,H−1 . For τ=0 , the residual reduces to Sk−1∣k−1rs=∣Ftk−1−Ftk∣ . The resulting error terms Sk+τ∣krs are then aggregated into the conformal score set S .
Mobile Robot
Quadrotor
Distribution
MC
MCA
MCA-CQR(ours)
MC
MCA
MCA-CQR(ours)
Uniform
0
100
100
45
100
100
%Succ
Gaussian
0
100
100
41
100
100
Time Varying
0
100
100
40
100
100
Uniform
-0.026
0.027
0.008
-0.056
0.102
0.054
minh
Gaussian
-0.128
0.032
0.006
-0.054
0.086
0.057
TABLE I: Quantitative experiment results under different distributions and different robot platforms
Fig. 3: Simulations for a mobile robot navigating in a single obstacle scenario under different noise distributions with different methods. γ=0.9 and α=0.05 are adopted. Our proposed framework (MCA-CQR) not only achieves safe navigation across different motion noise distributions but also provides a smaller safety margin than the MCA.
Fig. 4: Simulations for a planar quadrotor navigating in a single obstacle scenario under variant noise distributions with different methods. γ=0.9 and α=0.05 are adopted. Our proposed framework (MCA-CQR) not only achieves safe navigation across different motion noise distributions but also provides a smaller safety margin than the MCA.
Fig. 5: Parameter analysis with MC [ 22 ] and MCA-CQR(ours) under different MPC horizon H and the γ in CBF. This is conducted under the mobile robot platform. MC collides with the obstacle across different horizons, even with a small γ , whereas our method can achieve a collision-free path. In this simulation, we apply the Gaussian distribution or the uniform distribution randomly to ϵk at every step k .
Fig. 6: Simulation for a planar quadrotor to track a circle while avoiding the obstacle. MC, MCA, and MCA-CQR are all applied. Results show that MC collides with the obstacle, whereas the uncertainty-aware method proposed in this paper (MCA-CQR) not only guarantees safety but also achieves a smaller safety margin than MCA under the residual score function.
Fig. 7: Simulations for a quadrotor to track a figure eight while avoiding the obstacle under the time-varying motion noise. γ=0.9 and α=0.05 are adopted. Our proposed framework (MCA-CQR) not only guarantees safety but also provides a small safety margin with respect to the MCA under the residual score function.
Safety-critical control of uncertain, adaptive systems often relies on conservative, worst-case uncertainty bounds that limit closed-loop performance. Online conformal prediction is a powerful data-driven method for quantifying uncertainty when truth values of predicted outputs are revealed online; however, for systems that adapt the dynamics without measurements of the state derivatives, standard online conformal prediction is insufficient to quantify the model uncertainty. We propose Staggered Integral Online Conformal Prediction (SI-OCP), an algorithm utilizing an integral score function to quantify the lumped effect of disturbance and learning error. This approach provides long-run coverage guarantees, resulting in long-run safety when synthesized with safety-critical controllers, including robust tube model predictive control. Finally, we validate the proposed approach through a numerical simulation of an all-layer deep neural network (DNN) adaptive quadcopter using robust tube (MPC), highlighting the applicability of our method to complex learning parameterizations and control strategies.
Daniel M. Cherenson, Dimitra Panagou
Robotics Department, University of Michigan, Ann Arbor, MI, USA
Safe navigation in dynamic and uncertain environments often relies on accurate estimation of, or assumptions about, the true underlying uncertainty. However, accurately characterizing the true uncertainty distribution is often difficult due to limited data or imperfect information. An incorrect understanding of the uncertainty and its associated risk may lead to dangerous decisions even under high levels of risk aversion. To address this issue, we propose a risk-aware model predictive control (RA-MPC) framework that incorporates prediction sets to guarantee risk control below a user-specified threshold without requiring assumptions about the underlying uncertainty distribution. To generate the prediction sets, we develop a distribution-free risk quantification framework that extends conformal risk control (CRC) to general spectral risk measures. We then show that incorporating the prediction sets into the MPC framework provides statistical safety guarantees in terms of spectral risk constraint satisfaction even under uncertainty misspecification. We validate the proposed framework in simulated vehicle obstacle avoidance scenarios, demonstrating improved safety and reduced solve time compared to a baseline RA-MPC framework.
Junsik Eom, Tulga Ersal
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109
Many safety-critical control systems operate under latent uncertainty that sensors cannot resolve at decision time. Such uncertainty, arising from unknown physical properties, disturbances, or unobserved geometry, affects dynamics, task feasibility, and safety margins. Standard methods optimize expected performance and offer limited protection against rare but severe outcomes, while robust formulations treat uncertainty conservatively without exploiting its probabilistic structure. We consider systems with measured state and an unknown, time-invariant parameter represented by a belief distribution. We propose a risk-sensitive belief-space Model Predictive Path Integral (MPPI) controller that plans under this belief, regularizes performance using Conditional Value-at-Risk (CVaR), and imposes a CVaR constraint on a trajectory safety margin over the horizon. For the exact risk-constrained formulation underlying this controller, we establish three properties: (1) the CVaR constraint implies a probabilistic safety guarantee, (2) the controller recovers the risk-neutral optimum as the objective risk weight tends to zero, and (3) a union-bound argument extends the per-horizon guarantee to cumulative safety over repeated solves. In contact-rich MuJoCo simulations of vision-guided dexterous stowing, where a manipulator inserts a grasped object into an occupied slot with pose uncertainty exceeding prescribed lateral clearance requirements, our method achieves 82% success with zero contact violations at high risk aversion, compared with 55% and 50% for a risk-neutral configuration and a chance-constrained baseline, both of which incur nonzero exterior contact forces. Project page: https://clintonenwerem.com/belief-cvar-mppi/.
Clinton Enwerem, John S. Baras, Calin Belta
Institute for Systems Research, University of Maryland, College Park, MD, USA