Distance and Collision Probability Estimation from Gaussian Surface Models (2402.00186v2)

Abstract: This paper describes continuous-space methodologies to estimate the collision probability, Euclidean distance and gradient between an ellipsoidal robot model and an environment surface modeled as a set of Gaussian distributions. Continuous-space collision probability estimation is critical for uncertainty-aware motion planning. Most collision detection and avoidance approaches assume the robot is modeled as a sphere, but ellipsoidal representations provide tighter approximations and enable navigation in cluttered and narrow spaces. State-of-the-art methods derive the Euclidean distance and gradient by processing raw point clouds, which is computationally expensive for large workspaces. Recent advances in Gaussian surface modeling (e.g. mixture models, splatting) enable compressed and high-fidelity surface representations. Few methods exist to estimate continuous-space occupancy from such models. They require Gaussians to model free space and are unable to estimate the collision probability, Euclidean distance and gradient for an ellipsoidal robot. The proposed methods bridge this gap by extending prior work in ellipsoid-to-ellipsoid Euclidean distance and collision probability estimation to Gaussian surface models. A geometric blending approach is also proposed to improve collision probability estimation. The approaches are evaluated with numerical 2D and 3D experiments using real-world point cloud data. Methods for efficient calculation of these quantities are demonstrated to execute within a few microseconds per ellipsoid pair using a single-thread on low-power CPUs of modern embedded computers

• The paper presents a novel methodology to compute the closest Euclidean distance, gradient, and collision probability between an ellipsoidal robot and a Gaussian-modeled environment.
• It refines traditional spherical approximations by extending optimization-based methods for numerically stable, microsecond-level performance on embedded systems.
• Empirical evaluations demonstrate up to a tenfold improvement in distance estimation accuracy, enhancing precision in uncertainty-aware motion planning.

Distance and Collision Probability Estimation from Gaussian Surface Models

The paper authored by Kshitij Goel and Wennie Tabib presents novel methodologies for estimating collision probability, Euclidean distance, and gradient between an ellipsoidal robot model and an environment surface modeled using Gaussian distributions. These methods aim to address limitations in collision detection and motion planning approaches that utilize spherical robot approximations by providing a tighter fit with ellipsoidal representations, which are advantageous in cluttered and narrow spaces.

Methodological Advancement

The core contribution of this work lies in its ability to compute:

1. The closest Euclidean distance between the robot body and the environment surface.
2. The gradient of this distance.
3. An upper bound on collision probability, given uncertainty in the robot's position.

The proposed techniques leverage Gaussian surface models (GSMs) to represent the environment. Unlike traditional methods that create spatially discretized maps, leading to increased computational burden, Gaussian process-based approaches allow for continuous-space distance and gradient estimation. However, existing methods, while effective for spherical approximations, do not extend naturally to ellipsoidal robots.

Computational Implementation and Numerical Stability

The paper extends Rimon and Boyd's optimization-based method for ellipsoid-to-ellipsoid distance estimation, transforming it to avoid explicit matrix inversions and simplify its integration into robotic systems. This extension provides numerically stable distance and gradient estimations, crucial for large workspaces where computational efficiency is paramount.

In terms of computational load, the proposed methods can operate in a matter of microseconds on single-threaded CPUs even within low-power embedded systems, making them suitable for real-time robotics applications. This efficiency is achieved through optimized C++ implementations that capitalize on the Eigen library for linear algebra operations.

Practical and Theoretical Implications

Practically, the proposed methods enhance the precision of robot navigation in complex environments by improving distance estimation accuracy when compared to state-of-the-art methods. The empirical evaluations demonstrate that the proposed methods outperform existing Gaussian process-based approaches in terms of both distance field accuracy and computation of gradients. Notably, in scenarios where the environment is represented by a Gaussian mixture model (GMM), the proposed technique achieved up to a tenfold increase in distance estimation accuracy.

Theoretically, these methods fill a critical gap in motion planning under uncertainty, particularly in enabling continuous-space collision probability calculations. The blending approach proposed for computing collision probabilities results in smoother probability fields, which are beneficial for uncertainty-aware motion planning.

Future Directions in AI

The research opens several avenues for future exploration. Notably, the extension of these methods to account for orientation spaces within special orthogonal groups, such as SO(2) and SO(3), would be a significant advancement. Additionally, integrating these methods with scalable local submap extraction or spatial partitioning data structures to handle larger environments efficiently would enhance practical applications.

Another interesting development could involve the deployment of concurrent or vectorized implementations of the eigenvalue problems central to these methods, further reducing computational time and enabling real-time applications in dynamic environments. Moreover, exploring the use of deep learning to improve the fitting of GSMs to point cloud data, potentially leveraging neural implicit representations, could offer an intriguing synergy between classical geometric techniques and modern neural network models.

Conclusion

The methodologies proposed by Kshitij Goel and Wennie Tabib represent a noteworthy advancement in the field of robotics, particularly for motion planning and collision avoidance using ellipsoidal robots. By addressing the limitations of existing approaches and proposing computationally efficient solutions, the research provides valuable tools for the robotics community, facilitating safer and more accurate navigation in complex and cluttered environments. Future work in extending these methods and integrating them with advanced mapping techniques promises further enhancements in robotic autonomy and intelligence.