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

The paper introduces CaPTURe, a geometry-aware conformal prediction algorithm that uses particle-based models to generate probabilistically valid, contact-aware uncertainty regions for robot configurations, thereby significantly improving task success rates in both contact-rich and contactless scenarios compared to existing baselines.

Original authors: Luís Marques, Kristian Popov, Dmitry Berenson

Published 2026-08-11

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Luís Marques, Kristian Popov, Dmitry Berenson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are teaching a robot to play a game of "keep-away" with a marble in a maze, or to carefully screw a tiny peg into a hole. The robot has a brain (a computer model) that tries to guess where the marble or peg will be next. But here's the catch: the real world is messy. The marble might bounce off a wall in a way the robot didn't expect, or the peg might get stuck because the hole is slightly smaller than the robot thinks. This is called uncertainty.

In the world of robotics, there are two main types of this messiness. First, there's random noise (like a sudden gust of wind or a slippery floor) that happens even if the robot's brain is perfect. Second, there's model mismatch, which is when the robot's brain is just wrong because it learned from too few examples or made a bad guess about how physics works. To keep robots safe, engineers need to know not just where the robot thinks it will go, but how sure it is. If the robot is too confident and wrong, it might crash. If it's too scared and unsure, it might just freeze and do nothing.

The tricky part is that when a robot touches something—like a wall or a peg—the rules of the game change instantly. The robot might slide along a wall (one dimension), get stuck in a corner (zero dimensions), or fly through the air (two or three dimensions). These sudden shifts make it incredibly hard to draw a "safety bubble" around where the robot might end up. Traditional methods often draw a simple, round bubble (like a circle or a ball) to represent this uncertainty. But if the robot is sliding along a wall, a round bubble is useless because it includes all the empty space behind the wall where the robot physically cannot go. This paper introduces a new way to draw these safety bubbles that respects the shape of the world, ensuring the robot knows exactly how much room it has to maneuver without crashing.

The Paper: CaPTURe – A Shape-Shifting Safety Net

The authors of this paper, Luís Marques, Kristian Popov, and Dmitry Berenson from the University of Michigan, propose a new method called CaPTURe (Calibrated Particle-sets for Trans-dimensional Uncertainty Representation). Think of CaPTURe as a super-smart, shape-shifting safety net that the robot uses to predict its future.

Instead of guessing a single future spot, the robot runs a simulation thousands of times in its head, creating a cloud of "what-if" scenarios called particles. Imagine throwing a handful of glitter into the air; the glitter represents all the possible places the robot could end up. The problem is, how do you draw a boundary around that glitter that is both tight (so the robot doesn't get too conservative) and safe (so it never misses a real possibility)?

CaPTURe solves this by realizing that the robot's world is stratified, which is a fancy word for "layered."

Layer 1: Free space (the robot is flying through the air).
Layer 2: Sliding on a wall (the robot is constrained to a line).
Layer 3: Stuck in a corner (the robot is stuck at a single point).

Old methods tried to draw one giant, round bubble around the glitter, ignoring these layers. This meant the safety bubble often included impossible places (like "inside the wall"), making the robot overly cautious or, worse, dangerously optimistic. CaPTURe, however, builds a separate, custom-shaped safety net for each layer. It asks: "If the robot ends up sliding on the wall, what does the safety net look like? If it's in free space, what does it look like?"

To make sure these nets are actually safe, the authors use a mathematical trick called Conformal Prediction. Imagine you are a teacher grading a test. You don't just guess what score a student will get; you look at how the student performed on a practice test (the "calibration dataset") to set a fair passing line. CaPTURe does this for the robot. It looks at a dataset of past moves to figure out exactly how much "wiggle room" to give the robot in each specific situation. It guarantees that if you set the safety level to 90%, the robot will actually be inside its safety net 90% of the time, no matter how messy the physics get.

The Experiments: Marbles and Pegs

The team tested CaPTURe in two very different, challenging simulations to see if it actually worked better than existing methods.

1. The Marble Labyrinth
They simulated a robot controlling a tilting board to guide a marble through a maze filled with pits. The marble has to slide along walls and navigate tight corners.

The Challenge: If the robot misjudges how fast the marble is moving, it might fall into a pit. The uncertainty changes drastically depending on whether the marble is in open space or sliding against a wall.
The Result: CaPTURe successfully navigated the maze with a 90% success rate in the hardest sections, significantly outperforming other methods. The other methods either got too conservative (refusing to move because their safety bubbles were too big) or too optimistic (falling into pits because their bubbles didn't account for the walls). CaPTURe's ability to change the shape of its safety net based on the marble's contact with the walls allowed it to take risks where it was safe and be careful where it wasn't.

2. The Peg-in-Hole Task
They also tested a robot arm trying to insert a round peg into a hole with a very tight tolerance (the peg is 7.986 mm and the hole is 9.000 mm).

The Challenge: This is a classic assembly task. If the robot is slightly off, the peg gets stuck or jams. The uncertainty here is "trans-dimensional," meaning the peg might be floating in 3D space, touching the edge of the hole (2D), or touching a corner (1D).
The Result: In this simulation, CaPTURe achieved a 78% success rate, which was much higher than the next best method (which only got 48%). The other methods struggled because they couldn't handle the sudden changes in the peg's movement when it touched the hole. CaPTURe's "contact-aware" safety nets allowed the robot to adjust its strategy the moment it felt a touch, leading to smoother and more successful insertions.

What This Means

The paper suggests that by acknowledging that robots move differently when they are touching things versus when they are flying through the air, we can make them much safer and more efficient. The authors show that you don't need a perfect model of the world to do this; you just need a good way to calibrate your guesses based on past data.

While these results are from simulations and not yet tested on real-world robots in a factory, the math proves that the method works. The authors found that their approach is particularly good at handling "model mismatch"—situations where the robot's internal model of physics is slightly wrong. By using these custom-shaped, layer-specific safety nets, CaPTURe helps robots navigate the messy, contact-filled real world with a level of confidence that was previously out of reach. It's a step toward robots that don't just guess where they are going, but truly understand the shape of their own uncertainty.

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

Problem Statement

Deploying autonomous robots in contact-rich environments (e.g., cluttered navigation, assembly tasks) requires reasoning about how uncertainty evolves across different interaction regimes. State and action uncertainty can change drastically when a robot transitions from free-space motion to contact with obstacles. These transitions often induce multimodal, discontinuous, and trans-dimensional uncertainty distributions, where the future configuration may lie on manifolds of lower intrinsic dimension (e.g., sliding along a wall) compared to the full configuration space (C-space).

Standard uncertainty representations, such as Gaussian distributions, often fail to capture these complex post-contact dynamics. Furthermore, analytical or learned motion models are frequently inaccurate due to model mismatch, limited data, or unmodeled physical effects (e.g., stiction, friction). This inaccuracy can lead to unsafe executions or task failure. While Conformal Prediction (CP) offers distribution-free probabilistic guarantees, existing robotic applications often provide only global uncertainty bounds that do not adapt to specific contact regimes or respect the geometric constraints of the C-space, potentially generating prediction regions that include physically infeasible configurations.

The core challenge addressed is constructing probabilistically valid prediction regions for future robot configurations that:

Adapt to the robot's state, action, and contact mode (stratum).
Respect the stratified structure of the C-space (avoiding infeasible regions).
Provide finite-sample coverage guarantees despite significant model mismatch and aleatoric uncertainty.

Methodology: CaPTURe

The authors propose Calibrated Particle-sets for Trans-dimensional Uncertainty Representation (CaPTURe), a geometry-aware algorithm that combines particle-based dynamics prediction with Mondrian Conformal Prediction.

1. Stratified Configuration Space

The method models the feasible C-space ( $C_{feas}$ ) as a stratified space, defined as a disjoint union of smooth submanifolds (strata) $S_m$ of differing intrinsic dimensions (e.g., 2D free space, 1D edge contact, 0D corner contact). A known indexer function $T$ maps any configuration to its corresponding stratum index.

2. Particle-Based Predictive Model

CaPTURe utilizes an approximate dynamics model $\hat{f}$ (of arbitrary fidelity) to generate a set of $L$ predictive particles $\hat{Y}_i$ representing the conditional distribution of the next configuration given the current state-action pair. This particle-based approach naturally captures multimodality and allows uncertainty to collapse onto lower-dimensional manifolds.

3. Mondrian Conformal Calibration

To achieve adaptive, contact-aware guarantees, the method extends Mondrian Conformal Prediction:

Grouping: Examples are partitioned into groups based on the current state-action pair and the future stratum index $T(Y)$ . This grouping function $g(X, T(Y))$ is learned using a regression decision tree (DTree) trained on a subset of the calibration data to minimize variance in nonconformity scores.
Nonconformity Score: The score $r$ is defined as the distance from the true next configuration $Y_i$ to its $k$ -nearest neighbor (kNN) among the predicted particles $\hat{Y}_i$ . This score serves as a proxy for local particle density.
Thresholding: For each group $j$ (leaf node of the DTree), a conformal threshold $\hat{q}_j$ is computed independently using a holdout subset of the calibration data. This ensures that the coverage guarantee holds conditionally within each stratum and state-action regime.

4. Prediction Region Construction

During online planning (e.g., Model Predictive Control), for a given state-action pair, the algorithm:

Queries the DTree to identify the group for each candidate future stratum.
Constructs a prediction region $\hat{C}^{(m)}$ within each stratum $S_m$ containing configurations with nonconformity scores below the group-specific threshold $\hat{q}_j$ .
Returns the union of these per-stratum regions as the final prediction set $\hat{C}$ .

This construction guarantees that the prediction region contains only feasible configurations (by restricting the search to valid strata) and satisfies group-conditional coverage guarantees.

Key Contributions

Algorithm: An algorithm for constructing contact-aware, one-step trans-dimensional prediction regions that are guaranteed to contain the unknown future configuration with at least a user-specified probability, despite epistemic and aleatoric uncertainty.
Theoretical Guarantees: A proof and numerical validation of stratum-aware group-conditional guarantees using an approximate dynamics model of arbitrary fidelity and a finite calibration set. The method ensures that prediction regions do not extend into infeasible C-space regions.
Empirical Validation: Simulation experiments on two distinct tasks demonstrating the utility of the approach for contact-rich motion planning under significant uncertainty.

Experimental Results

The authors evaluated CaPTURe against four baselines: an uncalibrated particle planner (ParticleNoCP), a local CP method with Gaussian assumptions (LUCCa), standard Probabilistic Conformal Prediction (PCP), and an ablation of CaPTURe without stratum labeling.

Task 1: Marble Labyrinth Control

Setup: Controlling a marble in a 2D maze with tight clearances and pits.
Findings:
- CaPTURe achieved the user-specified 90% coverage requirement across all strata (Free Space, Edge, Corner).
- Baselines like LUCCa and PCP significantly undercovered in free space (83.4% and 76.4% respectively) while being overconservative in lower-dimensional strata.
- CaPTURe achieved a 90% task success rate in the "Center" section, significantly outperforming ParticleNoCP (53%) and PCP (3%).
- The method produced tighter, more volume-efficient prediction regions compared to Gaussian-based baselines.

Task 2: Tight-Tolerance Peg-in-Hole Insertion

Setup: A 7-DoF manipulator inserting a peg into a low-tolerance hole under stochastic disturbances and model mismatch (multiplicative noise).
Findings:
- CaPTURe achieved 90.4% empirical coverage across aggregate interactions, with sufficient coverage in Free Space, Point Contact, and Edge Contact regimes.
- Baselines (LUCCa, PCP) undercovered in Free Space (approx. 83-84%).
- In planning, CaPTURe achieved a 78% success rate, compared to 48% for PCP and LUCCa, and 14% for the uncalibrated baseline.
- The method reduced the average number of steps to completion (28.7 steps vs. 39.9 for the baseline).

Significance and Claims

The paper claims that CaPTURe provides a practical framework for safe and efficient motion planning in contact-rich environments where uncertainty is non-Gaussian and trans-dimensional. By explicitly modeling the stratified nature of the configuration space and using particle-based conformal prediction, the method generates feasible prediction regions that adapt to local contact regimes.

The authors emphasize that while the theoretical guarantees are strictly single-step, the recursive application of these calibrated regions within a Model Predictive Control (MPC) framework yields significant improvements in closed-loop task success rates and efficiency. The work demonstrates that accounting for the geometric structure of contact interactions (strata) is critical for accurate uncertainty quantification, preventing the over-conservatism or unsafe under-coverage observed in existing global or parametric approaches.

The paper concludes that treating robot motion uncertainty as dependent on state, action, and configuration stratum enables the creation of informative, trans-dimensional uncertainty representations essential for reliable autonomous systems operating in complex, real-world environments.