Particle-Based Conformal Prediction for Contact-Aware Uncertainty Calibration in Stratified Configuration Spaces

Particle-Based Conformal Prediction for Contact-Aware Uncertainty Calibration in Stratified Configuration Spaces by Luís Marques, Kristian Popov, and Dmitry Berenson (Department of Robotics and Department of Aerospace Engineering, University of Michigan). Published in Proceedings of Machine Learning Research 329:1–31, 2026, Conformal and Probabilistic Prediction with Applications.

The paper addresses the challenge of reliable uncertainty representation for autonomous systems that interact with their environment through contact. The authors state: "Reliable uncertainty representation is essential for deploying autonomous systems that interact with their environment, as robots must reason about how uncertainty arising from both stochasticity and model mismatch is impacted by contacts with obstacles (e.g., when navigating through a cluttered environment or inserting a part into an assembly)."

Key challenges identified include:

• Analytical or learned motion models are often inaccurate due to limited data, simplifying assumptions, or unmodeled effects

• When a robot contacts an obstacle, the distribution of future configurations can become multimodal or disjoint, or lie along manifolds of lower intrinsic dimension than the space of possible robot configurations

• Contact interactions can induce multimodal, discontinuous, and trans-dimensional uncertainty distributions over future robot configurations

The authors propose Calibrated Particle-sets for Trans-dimensional Uncertainty Representation (CaPTURe), described as a geometry-aware, conformal prediction-based algorithm that generates probabilistically valid prediction regions of the unknown future system configuration using particle-based models of arbitrary fidelity.

The method combines:

1. Mondrian Conformal Prediction - partitioning examples into groups and calibrating each independently

2. Probabilistic Conformal Prediction (PCP) - constructing prediction regions using particle-based predictive models

The paper lists three main contributions:

1. "An algorithm for constructing contact-aware, one-step trans-dimensional prediction regions guaranteed to contain the unknown future robot configuration with at least a user-specified probability, despite epistemic and aleatoric uncertainty."

2. A proof and numerical validation of our stratum-aware group-conditional guarantees using an approximate dynamics model of arbitrary fidelity and a finite calibration set.

3. Simulation experiments of a marble labyrinth control task and a manipulator peg-insertion task, demonstrating the utility of our approach for contact-rich motion planning under significant uncertainty.

The paper treats the robot's feasible configuration space as a stratified space: A finite stratified space Cfeas is a disjoint union Cfeas:= ∪m∈M Sm, where each stratum is denoted by Sm, and T: Cfeas → M maps each configuration to the index m of the stratum containing it.

The method defines a grouping function g: X × M → 1,..., J that partitions examples according to the prediction-time information X and the future C-space stratum T(Y). This allows calibration to adapt across C-space strata with distinct dynamics and uncertainty structures.

The score function is defined as the kNN-th nearest neighbor distance: r(f̂(Xi), Yi):= kNN kNN(Yi, Ŷi), where Yi is the true next configuration, Ŷi is the set of L propagated particles, and kNN kNN denotes the distance to the kNN-th closest particle.

The implementation uses:

• A regression decision tree (CART) fitted on augmented inputs (Xi, T(Yi)) with scores Ri as targets

• The tree partitions the joint X × M space to find regions of approximately constant model error

• Separate conformal thresholds q̂j are computed per leaf node using SplitCP

• The calibration dataset is split into two parts: one for fitting the DTree, one for computing conformal thresholds

The paper proves:

• Theorem 5: Group-conditional coverage guarantee - P Yn+1 ∈ Ĉ(Xn+1) g(Xn+1, T(Yn+1)) = j ≥ 1 − α for every group j

• Corollary 6: Marginal coverage by union of output strata - the union of group-specific prediction regions satisfies marginal coverage

The authors note: "We emphasize that the coverage guarantees from Theorem 5 apply only to single-step prediction. Providing formal multistep coverage guarantees is not trivial because Ĉ at later planning steps depends recursively on predicted states, particles, and Ĉ earlier in the horizon."

A planar marble control environment inspired by the BRIO labyrinth toy was simulated. The system state includes marble position, velocity, and plate inclination angles. The approximate dynamics model differs from the true system through a multiplicative dynamics term introducing considerable model mismatch.

Coverage Results: Over 427,680 test cases, CaPTURe achieved the user-specified 90% coverage in all strata (Free Space, Edge, Corner), while baselines significantly undercovered in free space (e.g., PCP achieved only 76.4% coverage in free space) while being overconservative in lower-dimensional strata.

Planning Results: CaPTURe achieved up to 90% success rate across maze sections, compared to 53% for ParticleNoCP, 3% for LUCCa, and 57% for PCP in the Center section. The paper reports up to a 30% absolute improvement in task success rate over the best baseline.

A 7-DoF Franka Panda manipulator inserts a cylindrical peg (7.986 mm diameter) into a 9.000 mm-diameter hole. The peg configuration lies in SE(2), and the C-space is stratified using binary contact fingerprints based on peg vertices and edges.

Coverage Results: Over 90,000 held-out transitions, CaPTURe achieved sufficient coverage in all interaction regimes (Free Space, Point Contact, Edge Contact), while baselines undercovered in free space (LUCCa: 83.8%, PCP: 83.1%, Ablation: 84.9%).

Planning Results: CaPTURe achieved 78% success rate compared to 14% for ParticleNoCP, 48% for PCP, 48% for LUCCa, and 48% for the ablation without stratum labels.

1. Our calibration procedure captures how motion uncertainty varies between contact-rich and contactless motions, leading to sufficient coverage in both cases.

2. Both variants of our method achieve the 1 − α = 0.9 coverage requirement per stratum, with the kNN = 8 variant being slightly more volume-efficient.

3. The two CaPTURe variants are the only methods that achieve both marginal and sufficient coverage within each reported contact category.

4. The results indicate that reliable uncertainty calibration can improve task success in tight-tolerance settings under significant model mismatch and aleatoric disturbances.

The authors acknowledge: "Our guarantees are single-step and, although we observe improved closed-loop performance, we do not prove multistep closed-loop coverage. CaPTURe assumes knowledge of the environmental constraints and access to a correct stratum indexer, since geometric uncertainty could produce prediction regions containing infeasible configurations."

The paper concludes: "We proposed a state-action-stratum-aware conformal prediction algorithm to construct probabilistically valid next-configuration prediction regions in stratified configuration spaces. Our approach combines particle-based dynamics predictors with a Mondrian-based adaptive calibration to produce uncertainty sets that can capture multimodal, discontinuous, and trans-dimensional robotic motion uncertainty and do not include infeasible regions of the robot's configuration space."

Based on this paper, I can improve AI systems in the following ways:

1. Contact-Aware Uncertainty Calibration for Motion Planning

• Implement a stratified conformal prediction module that partitions the configuration space into contact-regime strata (free space, edge contact, corner contact) and calibrates uncertainty bounds independently per stratum.

• The improved system can generate prediction regions that remain valid (≥ user-specified coverage) even when robot dynamics change abruptly due to collisions, preventing overconfident plans near obstacles.

2. Trans-Dimensional Uncertainty Handling

• Use particle-based predictive models with kNN nonconformity scores to represent multimodal, discontinuous, or lower-dimensional uncertainty distributions (e.g., when a peg can contact a hole at multiple discrete points).

• The improved system can output prediction sets that are not constrained to a single Gaussian or unimodal shape, enabling safer planning in tight-tolerance assembly or navigation in cluttered environments.

3. Group-Conditional Calibration via Mondrian Conformal Prediction

• Integrate a regression decision tree that groups calibration data by both input features and future stratum labels, then compute separate conformal thresholds per leaf.

• The improved system can adapt its uncertainty bounds to regions of the state space where model error is systematically different (e.g., high error near walls, low error in open space), avoiding both under-coverage and over-conservatism.

4. Single-Step Guarantee with Recursive Planning Integration

• Use the single-step calibrated prediction regions as constraints in a model predictive control (MPC) or sampling-based planner, where each step's region is recomputed from the current particle distribution.

• The improved system can achieve higher task success rates (e.g., 78% vs 48% for baselines in peg insertion) by avoiding plans that rely on uncertain future states that fall outside calibrated regions, even without formal multi-step guarantees.

5. Stratum-Aware Planning with Infeasible-Region Exclusion

• Constrain prediction regions to lie within the feasible stratified configuration space, using the known stratum indexer to clip or project particles.

• The improved system can generate plans that never propose configurations in infeasible regions (e.g., inside obstacles), which is critical for contact-rich tasks where naive uncertainty sets might include impossible states.

6. Adaptive Calibration with Finite Data

• Use a split conformal approach where one part of the calibration set fits the grouping tree and another computes thresholds, enabling reliable guarantees with limited data.

• The improved system can be deployed in new environments with modest calibration datasets, still achieving per-stratum coverage guarantees despite model mismatch and aleatoric noise.

7. Closed-Loop Performance Improvement via Uncertainty-Aware Cost Functions

• Incorporate the calibrated prediction region size or coverage status into the planning cost (e.g., penalize trajectories that pass through high-uncertainty strata).

• The improved system can automatically trade off between exploration and safety, leading to up to 30% absolute improvement in task success in contact-rich manipulation and navigation benchmarks.

Reliable uncertainty representation is essential for deploying autonomous systems that interact with their environment, as robots must reason about how uncertainty arising from both stochasticity and model mismatch is impacted by contacts with obstacles (e.g., when navigating through a cluttered environment or inserting a part into an assembly). We propose Calibrated Particle-sets for Trans-dimensional Uncertainty Representation (CaPTURe), a geometry-aware, conformal prediction-based algorithm that generates probabilistically valid prediction regions of the unknown future system configuration using particle-based models of arbitrary fidelity. While calibrated uncertainty predictions are essential for safe and efficient planning, analytical or learned motion models are often inaccurate - due to limited data, simplifying assumptions, unmodeled effects, etc. - which can lead to unsafe executions or task failure. Additionally, when a robot contacts an obstacle, the distribution of its future configurations can become multimodal or disjoint, or lie along manifolds of lower intrinsic dimension than the space of possible robot configurations. Our method uses a calibration dataset of system transitions to locally calibrate motion uncertainty estimates, constructing regions guaranteed to contain the future robot configuration at a user-set probability. Our calibration procedure captures how motion uncertainty varies between contact-rich and contactless motions, leading to sufficient coverage in both cases. We evaluate our method on two simulated planning tasks: controlling a marble around a labyrinth and performing tight-tolerance peg-in-hole insertion with a manipulator. Compared to relevant baselines, CaPTURe achieves the user-specified coverage requirement both in and out of contact and achieves up to a 30% absolute improvement in task success rate over the best baseline.