ch.alpine.ascona

Computational non-linear geometry demos in Java 17

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.

Integration

Specify repository and dependency of the owl library in the pom.xml file of your maven project:

<dependencies>
  <dependency>
    <groupId>ch.alpine</groupId>
    <artifactId>ascona</artifactId>
    <version>0.0.1</version>
  </dependency>
</dependencies>

<repositories>
  <repository>
    <id>ascona-mvn-repo</id>
    <url>https://raw.github.com/datahaki/ascona/mvn-repo/</url>
    <snapshots>
      <enabled>true</enabled>
      <updatePolicy>always</updatePolicy>
    </snapshots>
  </repository>
</repositories>

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