Predictable Robots: Calibrating Uncertainty for Safer Action

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Author: Denis Avetisyan

A new approach uses advanced mathematical tools to build more reliable predictions of robot movements, even with imperfect information about the environment.

The system leverages state transition data alongside an approximate dynamical model to establish a symmetry-aware probabilistic error bound on configuration predictions, enabling the computation of a calibrated prediction region $𝒞^q$ guaranteed to contain the true configuration resulting from a desired action $u_{des}$.
The system leverages state transition data alongside an approximate dynamical model to establish a symmetry-aware probabilistic error bound on configuration predictions, enabling the computation of a calibrated prediction region $𝒞^q$ guaranteed to contain the true configuration resulting from a desired action $u_{des}$.

This work introduces CLAPS, an algorithm leveraging Lie group theory and conformal prediction to quantify and calibrate uncertainty in robot configurations within nonholonomic systems.

Despite advances in robotics, guaranteeing safe and reliable operation under uncertainty remains a fundamental challenge. This is addressed in ‘Lies We Can Trust: Quantifying Action Uncertainty with Inaccurate Stochastic Dynamics through Conformalized Nonholonomic Lie Groups’, which introduces a novel framework for constructing calibrated prediction regions for robot configurations. By leveraging Lie group theory and conformal prediction, the proposed algorithm, CLAPS, provides probabilistic guarantees without strong assumptions about system dynamics or uncertainty sources. Could this symmetry-aware approach unlock more efficient and trustworthy robotic systems in complex, real-world environments?

The Dance of Prediction: Capturing the Essence of Movement

The ability to faithfully capture and foresee movement underpins a remarkable range of technologies, from the nuanced actions of robotic systems to the compelling realism of modern animation and the predictive power of complex simulations. For robotics, precise motion representation is critical for path planning, object manipulation, and safe interaction with dynamic environments. In animation and visual effects, realistic motion breathes life into digital characters and scenes, demanding algorithms that mimic the subtleties of physical movement. Similarly, simulations-whether modeling weather patterns, fluid dynamics, or biomechanical systems-rely on accurate motion representation to generate reliable and insightful predictions; a failure to do so can propagate errors and invalidate results, highlighting the foundational importance of this challenge across diverse scientific and creative domains.

Representing three-dimensional movement presents a considerable challenge for computational systems due to the inherent complexities of real-world motion and the uncertainties involved in perceiving it. Traditional techniques, often relying on predefined motion capture data or simplified kinematic models, frequently struggle to generalize to novel situations or adapt to unexpected disturbances. The continuous nature of movement, combined with the infinite degrees of freedom possible in $3D$ space, creates a high-dimensional data space that is difficult to model accurately. Furthermore, sensor noise, imperfect tracking, and the unpredictable behavior of dynamic systems introduce inherent uncertainties that can significantly degrade the performance of motion planning and control algorithms. This necessitates the development of more robust and adaptable methods capable of handling the complexities and uncertainties inherent in representing realistic motion.

The depiction of movement in computational systems relies heavily on representing transformations – changes in position and orientation – as rigid bodies. This necessitates a deep understanding of the mathematical framework underpinning these transformations, specifically utilizing concepts from linear algebra and group theory. A rigid transformation preserves distances and angles, demanding precise calculations using matrices – particularly rotation matrices and translation vectors – to accurately map points in space from one coordinate frame to another. Furthermore, composing multiple transformations requires careful attention to the order of operations, as matrix multiplication is not commutative. Failing to account for these mathematical nuances can lead to accumulated errors and unrealistic or unstable motion, particularly in complex simulations or robotic systems where precise control is paramount. Consequently, researchers are continually refining algorithms and data structures to efficiently and accurately represent and manipulate these transformations, ensuring the fidelity of simulated and real-world movements.

CLAPS constructs efficient, probabilistically guaranteed prediction regions in configuration space to reliably estimate future system states, leveraging robot symmetry to improve performance.
CLAPS constructs efficient, probabilistically guaranteed prediction regions in configuration space to reliably estimate future system states, leveraging robot symmetry to improve performance.

The Shadow of Uncertainty: Quantifying the Immeasurable

Accurate characterization of uncertainty is fundamental to the performance of motion prediction and control systems, particularly when operating within dynamic and unpredictable environments. Without a robust understanding of potential deviations from predicted trajectories, robotic systems cannot reliably plan safe and effective actions. Uncertainty arises from multiple sources including sensor noise, imperfect models of the environment, and the inherent stochasticity of physical systems. Quantifying this uncertainty allows for the development of more resilient controllers and planners, enabling robots to adapt to unforeseen circumstances and avoid collisions. Consequently, methods that improve the fidelity of uncertainty representation directly contribute to the safety and efficiency of robotic operation in complex, real-world scenarios.

Motion capture systems, while widely used for acquiring kinematic data, are inherently subject to inaccuracies. Sources of error include sensor noise arising from the electronic components and digitization of analog signals, limited sensor resolution which restricts the precision of recorded positions, occlusion where sensors lose tracking due to obstructions, and calibration errors affecting the coordinate system transformation. Furthermore, the accuracy of motion capture is impacted by the dynamic range and sampling rate of the sensors, and the inherent limitations of the tracking technology itself – optical, inertial, or magnetic – in capturing fast or complex movements. These limitations necessitate robust data processing and uncertainty modeling techniques to mitigate the effects of these errors and ensure the reliability of derived motion predictions and control algorithms.

Monte Carlo (MC) simulation is a computational technique used to model the propagation of uncertainty in dynamic systems. By running a motion model multiple times with randomly sampled initial conditions and parameters, MC methods generate a distribution of possible outcomes, effectively representing the range of plausible motions. This contrasts with deterministic models that provide a single predicted trajectory. Recent research, specifically the CLAPS system, has shown performance gains over standard MC simulations in representing this uncertainty. Evaluations in simulated environments demonstrate that CLAPS is capable of more accurately modeling the probabilistic nature of motion, yielding improved predictions and control compared to traditional calibrated baselines and direct implementations of MC particle methods.

Evaluation across 625 validation trials demonstrated that the CLAPS method consistently produced smaller configuration space (C-Space) volumes than all calibrated baseline methods. Furthermore, CLAPS achieved the highest average Intersection over Union (IoU) score when compared to Monte Carlo (MC) particle sets, indicating a more accurate representation of the underlying uncertainty in predicted motion. These results quantitatively validate CLAPS’s ability to provide a refined and reliable estimation of potential future states, outperforming established uncertainty propagation techniques in terms of both volume and overlap with probabilistic samples.

CLAPS effectively captures stochastic uncertainty in configuration space, providing more accurate coverage predictions than methods like SS EKF and InEKF, which struggle to meet the user-defined likelihood, and outperforms the overly-large regions generated by point prediction baselines.
CLAPS effectively captures stochastic uncertainty in configuration space, providing more accurate coverage predictions than methods like SS EKF and InEKF, which struggle to meet the user-defined likelihood, and outperforms the overly-large regions generated by point prediction baselines.

The Geometry of Motion: Navigating Non-Euclidean Realms

The configuration space of a rigid body, defined as the set of all possible positions and orientations, is frequently non-Euclidean due to the nature of representing rotations. While translational degrees of freedom exist in a Euclidean space, orientations are typically described using rotation matrices or quaternions, which form a non-Euclidean manifold – specifically, the special orthogonal group $SO(3)$ in three dimensions. This means that standard Euclidean distance metrics and averaging operations are not directly applicable to orientations; for instance, averaging two rotation matrices does not necessarily result in a valid rotation. Consequently, algorithms operating on configurations must account for the underlying non-Euclidean geometry to avoid errors in calculations such as state estimation or motion planning.

The Special Euclidean groups, $SE(2)$ and $SE(3)$, formally define rigid transformations in two and three dimensions, respectively. $SE(2)$ represents planar transformations comprising rotations and translations, parameterized by three degrees of freedom. $SE(3)$ extends this to three-dimensional space, incorporating three rotational and three translational degrees of freedom. Mathematically, $SE(n)$ is the group of $n \times n$ real matrices that preserve distances and orientations; it combines the special orthogonal group $SO(n)$ (rotations) with the vector space $\mathbb{R}^n$ (translations) via a semi-direct product. These groups are essential because they provide a consistent and mathematically rigorous way to represent the pose – position and orientation – of a rigid body, which is fundamental to robotics, computer vision, and related fields.

Accurate motion modeling and uncertainty propagation rely on the mathematical properties of the configuration space. Representing robot pose with groups like $SE(2)$ and $SE(3)$ ensures that transformations are properly composed and avoids singularities common in Euler angle representations. Using these groups allows for consistent and valid state estimation, crucial for tasks like localization and mapping. Furthermore, propagating uncertainty – represented as covariance matrices – requires operations consistent with the non-Euclidean geometry of these spaces; standard Kalman filter updates are invalid when applied directly to angular parameters represented as simple scalars, necessitating specialized formulations like those employing quaternions or Lie groups to maintain positive definite covariance matrices and prevent filter divergence.

Hardware testing of the CLAPS algorithm was conducted utilizing the MBot robotic platform to evaluate its performance in configuration space (C-Space). Results indicate that CLAPS achieved a 23% reduction in C-Space volume compared to the Simultaneous Localization and Mapping (SLAM) baseline consisting of Square Root Extended Kalman Filter (SS EKF) and Covariance Propagation (CP). This volumetric reduction suggests CLAPS can represent uncertainty more efficiently, potentially leading to improved path planning and reduced computational cost in robotic applications. The observed performance difference demonstrates a quantifiable practical benefit of implementing CLAPS over traditional SLAM approaches on a physical robotic system.

Transforming nonholonomic systems between state space and Lie group representations reveals how accelerations influence velocities and introduces cross-dimensional couplings-specifically, rotational uncertainty impacting positional uncertainty-through the Reconstruction equation and the ad* term.
Transforming nonholonomic systems between state space and Lie group representations reveals how accelerations influence velocities and introduces cross-dimensional couplings-specifically, rotational uncertainty impacting positional uncertainty-through the Reconstruction equation and the ad* term.

The pursuit of robust robotic systems, as detailed in this work, inherently acknowledges the inevitable decay of any predictive model. CLAPS, with its conformal prediction framework applied to nonholonomic Lie groups, doesn’t seek to eliminate uncertainty, but rather to gracefully account for it. This resonates with Vinton Cerf’s observation: “The Internet treats everyone the same.” Similarly, CLAPS treats all potential robot configurations with a calibrated level of skepticism, acknowledging that even the most sophisticated models are fallible. The algorithm doesn’t promise perfect foresight, but delivers a prediction region reflecting the true, potentially wide, range of possible outcomes, allowing for safer and more reliable operation over time. This isn’t merely about prediction accuracy; it’s about building systems that mature through acknowledging and adapting to their inherent limitations.

What Lies Ahead?

The pursuit of calibrated prediction, as demonstrated by CLAPS, isn’t about eliminating uncertainty-that’s an asymptotic ideal, a vanishing point on the horizon. Rather, it’s about acknowledging the inherent erosion within any dynamical system. Technical debt accumulates in the prediction space, much as sediment builds against a dam. This work offers a method to map that accumulation, to define regions where trust-or, more accurately, managed distrust-can be provisionally maintained. But the nonholonomic constraints considered here represent a simplification; real-world systems are riddled with asymmetries, unmodeled interactions, and the simple chaos of scale.

Future efforts will likely focus on extending these techniques to higher-dimensional, more complex spaces. The conformal prediction framework offers a degree of robustness, but its performance hinges on the quality of the underlying prediction model. Improving the efficiency of those models, and quantifying the rate of calibration decay, will be crucial. Uptime, in this context, becomes a rare phase of temporal harmony-a fleeting moment before the inevitable return to uncertainty.

Ultimately, the challenge isn’t to build systems that avoid failure, but to build systems that fail gracefully, and predictably. This requires a shift in perspective: from seeking absolute knowledge, to embracing the inevitability of incomplete information, and designing for resilience within that landscape. The question isn’t whether a system will eventually drift out of calibration-it’s how long that drift can be tolerated, and what safeguards can be put in place before it becomes catastrophic.

Original article: https://arxiv.org/pdf/2512.10294.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-15 02:31