Galilean Symmetry in Robotics

2025-10-12

AI Summary

Robotics has long lacked a systematic modeling framework for Galilean symmetry—the natural symmetry of inertial motion—while existing approaches predominantly focus on rigid-body symmetries without unifying them with core problems such as inertial navigation and kinematics. Method: This paper establishes the first Galilean-symmetry-theoretic framework for robotics, introducing the Galilean matrix Lie group to jointly represent Galilean reference frames and extended poses, thereby enabling geometrically consistent representations of both inertial and coordinate velocities. Leveraging Lie group and Lie algebra tools, it further models dynamics and temporal uncertainty under Earth’s rotating frame. Results: Evaluated on inertial navigation, robotic manipulator kinematics, and multi-sensor fusion tasks, the framework demonstrates strong physical interpretability, modeling concision, and adaptability to dynamic environments—introducing a new geometric modeling paradigm for robotics.

Technology Category

• Intelligent Robots: State Estimation
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Galilean symmetry is the natural symmetry of inertial motion that underpins Newtonian physics. Although rigid-body symmetry is one of the most established and fundamental tools in robotics, there appears to be no comparable treatment of Galilean symmetry for a robotics audience. In this paper, we present a robotics-tailored exposition of Galilean symmetry that leverages the community's familiarity with and understanding of rigid-body transformations and pose representations. Our approach contrasts with common treatments in the physics literature that introduce Galilean symmetry as a stepping stone to Einstein's relativity. A key insight is that the Galilean matrix Lie group can be used to describe two different pose representations, Galilean frames, that use inertial velocity in the state definition, and extended poses, that use coordinate velocity. We provide three examples where applying the Galilean matrix Lie-group algebra to robotics problems is straightforward and yields significant insights: inertial navigation above the rotating Earth, manipulator kinematics, and sensor data fusion under temporal uncertainty. We believe that the time is right for the robotics community to benefit from rediscovering and extending this classical material and applying it to modern problems.

Problem

• Introducing Galilean symmetry tailored for robotics applications
• Providing matrix Lie-group representations for inertial navigation
• Applying Galilean algebra to manipulator kinematics and sensor fusion

Innovation

• Uses Galilean matrix Lie group for pose representations
• Applies Galilean symmetry to robotics problems directly
• Provides examples in navigation and sensor fusion