ch.alpine.ascony

Computational non-linear geometry demo utils in Java 25

Student Projects

2019

• Oliver Brinkmann (MT): Averaging on Lie Groups: Applications of Geodesic Averages and Biinvariant Means
• Joel Gächter (MT): Subdivision-Based Clothoids in Autonomous Driving

Nearest Neighbors

R^2

Dubins

Clothoid

Visualization of Geodesic Averages

A geodesic average is the generalization of an affine combination from the Euclidean space to a non-linear space. A geodesic average consists of a nested binary averages. Generally, an affine combination does not have a unique expression as a geodesic average. Instead, several geodesic averages reduce to the same affine combination when applied in Euclidean space.

Contributors

Jan Hakenberg, Jonas Londschien, Yannik Nager, André Stoll, Joel Gaechter

Publications

• What lies in the shadows? Safe and computation-aware motion planning for autonomous vehicles using intent-aware dynamic shadow regions by Yannik Nager, Andrea Censi, and Emilio Frazzoli, video

References

• A Generalized Label Correcting Method for Optimal Kinodynamic Motion Planning by Brian Paden and Emilio Frazzoli, arXiv:1607.06966, video
• Sampling-based algorithms for optimal motion planning by Sertac Karaman and Emilio Frazzoli, IJRR11

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The library was developed with the following objectives in mind

• trajectory design for autonomous robots
• suitable for use in safety-critical real-time systems
• implementation of theoretical concepts with high level of abstraction

Curve Subdivision

Smoothing

Wachspress

Dubins path curvature

Features

• geodesics in Lie-groups and homogeneous spaces: Euclidean space R^n, special Euclidean group SE(2), hyperbolic half-plane H2, n-dimensional sphere S^n, ...
• parametric curves defined by control points in non-linear spaces: GeodesicBSplineFunction, ...
• non-linear smoothing of noisy localization data GeodesicCenterFilter
• Dubins path

Visualization of Geodesic Averages

A geodesic average is the generalization of an affine combination from the Euclidean space to a non-linear space. A geodesic average consists of a nested binary averages. Generally, an affine combination does not have a unique expression as a geodesic average. Instead, several geodesic averages reduce to the same affine combination when applied in Euclidean space.

Contributors

Jan Hakenberg, Oliver Brinkmann, Joel Gächter

Publications

• Curve Subdivision in SE(2) by Jan Hakenberg, viXra:1807.0463, video
• Smoothing using Geodesic Averages by Jan Hakenberg, viXra:1810.0283, video
• Curve Decimation in SE(2) and SE(3) by Jan Hakenberg, viXra:1909.0174

References

• Bi-invariant Means in Lie Groups. Application to Left-invariant Polyaffine Transformations. by Vincent Arsigny, Xavier Pennec, Nicholas Ayache
• Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups by Xavier Pennec, Vincent Arsigny
• Lie Groups for 2D and 3D Transformations by Ethan Eade
• Manifold-valued subdivision schemes based on geodesic inductive averaging by Nira Dyn, Nir Sharon
• Power Coordinates: A Geometric Construction of Barycentric Coordinates on Convex Polytopes by Max Budninskiy, Beibei Liu, Yiying Tong, Mathieu Desbrun