Dynamical System Modeling and Stability Investigation�DSMSI-2025

May 08-10, 2025, Kyiv, Ukraine

Coordinate free method for the identification of the plane curves and its applications to motion planning

Alexander Krasavin, Darya Alontseva, Albina Kadyroldina, Iurii Krak*, Dana Baishuak

D. Serikbayev East Kazakhstan Technical University

*Taras Shevchenko National University of Kyiv

​

Introduction

Problem:

Representing and identifying plane curves is crucial in various fields.

Limitations of coordinate-dependent methods: Sensitivity to translation, rotation, and scaling

Proposed Solution:

The angular characteristic function – a coordinate-free representation for continuous plane curves

Invariance under similarity transformations (translation, rotation, uniform scaling)

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Introduction

• Rapid advancements in machine learning and computer vision have led to widespread applications in image recognition and robot control
• Identifying graphical patterns requires effective feature extraction
• We propose a coordinate-free method for representing continuous plane curves using the angular characteristic function
• This representation is invariant under translations, rotations, and uniform scaling
• Applications include image recognition (handwritten characters) and motion planning (robotics)

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Objectives

• Define the angular characteristic function.
• Explore its key properties and mathematical foundations.
• Demonstrate its application in motion planning, with a focus on robotic manipulators.
• Highlight its use in image recognition and pattern analysis.

Dynamical System Modeling and Stability Investigation, DSMSI-2025

The angular characteristic function of a smooth plane curve

Dynamical System Modeling and Stability Investigation, DSMSI-2025

The Angular Characteristic Function of a parametrized plane curve

Dynamical System Modeling and Stability Investigation, DSMSI-2025

The Angular Characteristic Function of the Polyline

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Properties of angular characteristic function

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Application to Image Recognition: Feature Extraction

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Converting Digital Image To Planar Graph

Inverted black and white Image

Image Gradient Magnitude

Curvilinear structures detection

Planar graph of an image

Image of the handwritten symbol

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Chains of Planar Graph as Polylines

Extracted chain of a planar graph

Angular characteristic function of the corresponding polyline

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Examples of Chains and Corresponding Angular Characteristic Functions

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Optimal Approximation of Plane Curves by Polysegment Lines� ��

Polysegment line AB with three segments

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Application to Motion Planning: Polysegment Lines

• Motion planning in robotics often requires approximating trajectories with simpler primitives.
• A polysegment line is a sequence of line segments and circular arcs.
• A conjugate polysegment line has adjacent segments that are smoothly connected (conjugate circular arcs or a line segment conjugate with an arc).
• Robot manipulators often execute commands like "move" and "circle".
• Approximating smooth curves with conjugate polysegment lines avoids issues with infinite acceleration at sharp corners of polylines.

Dynamical System Modeling and Stability Investigation, DSMSI-2025

The angular characteristic function of a polysegment line is a continuous piecewise linear function.�

Polysegment line AB with three segments

Angular characteristic function of the polysegment curve AB

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Basic idea: Use of the piecewise linear approximation of the angular characteristic function

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Any angular characteristic function define a set of the plane curves

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Algorithm for Optimal Polysegment Curve Approximation

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Conclusion

• The coordinate-free angular characteristic function provides a powerful tool for representing and identifying plane curves
• Its invariance properties are beneficial for shape recognition
• It enables effective approximation of smooth curves with conjugate polysegment lines for smooth robot trajectories, avoiding infinite acceleration
• Experimental results (as a proof of concept) show high reliability in handwritten character recognition using this method
• Future prospects: Further research in optimal approximation algorithms and applications in various fields.

Acknowledgment

This research was funded by the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19679327 "Methods of machine learning for tasks of automatic control and inertial navigation of mobile robots")

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Thank you for your attention

Dynamical System Modeling and Stability Investigation, DSMSI-2025