Making Space for Time: The Special Galilean Group and Its Application to Some Robotics Problems

Group synchronization (GS) is the problem of estimating a set of $N$ unknown elements $g_1,\ldots, g_N \in \mathcal{G}$ in a group $\mathcal{G}$, given noisy measurements of a subset of their pairwise ratios $g_i^{-1} g_j$. GS problems lie at the core of many state estimation tasks in robotics and computer vision, including 3D vision, robotic mapping, inertial navigation, and molecular reconstruction. Unfortunately, GS problems are typically both high-dimensional and non-convex, and therefore hard to solve in general. In this paper, we present Fast-Sync, a fast linear approximation method for GS that is suitable for initializing local manifold-based optimizers or certifiable global methods. Our approach generalizes chordal initialization to arbitrary matrix Lie groups, and additionally proposes two new key algorithmic enhancements: we show how to exploit both the Kronecker-product structure in the problem data matrix and the topology of the synchronization graph to improve speed, scalability, and accuracy. Experimental evaluation across several GS tasks demonstrates that Fast-Sync provides high-quality initializations that enable local optimizers to efficiently recover globally optimal GS solutions, achieving high success rates even with considerable measurement noise.

The Python robotics ecosystem faces a challenge: while many libraries exist for rigid body transformations, few are both lightweight and mathematically strict. This paper introduces SE3Kit, a lightweight Python library efficient operations on the Special Euclidean Group SE(3) and the Special Orthogonal Group SO(3). Unlike established frameworks that require heavy dependencies (e.g., SpatialMath, PyPose) or general tools that lack robotics-specific features (e.g., SciPy), SE3Kit targets the gap between these extremes. It is designed for embedded deployment, rapid prototyping, and education while providing rigorous mathematical implementation. It provides a pure-Python, NumPy-only implementation of Lie Group operations, without the overhead of deep learning or other visualization software.

Many robotic systems allow independent control of position and orientation (pose), including omnidirectional aerial vehicles, underwater robots, and manipulator end-effectors. In many applications, these systems must follow a continuous sequence of poses, leading to either trajectory-tracking or path following formulations. Compared to trajectory-tracking, path following offers important practical advantages. In particular, we focus on the problem of path following on Lie groups. Considering the robots as rigid bodies moving in the 3D space, this path-following problem can be posed as a problem of designing guiding vector fields on the matrix Lie group SE(3). In this paper, we develop a general vector-field framework for path following on connected matrix Lie groups, of which SE(3) is a prominent special case. The proposed vector field guarantees convergence to a desired parametric curve from almost all initial conditions while ensuring continuous motion along the path. Furthermore, another interesting feature is that, as opposed to previous works, the control input is "minimal" in terms of representation and closer to the engineering application (e.g., the body twist in the case SE(3)). After establishing the general case, the framework is then specialized to SE(3), of special interest in robotics, yielding an efficient algorithm suitable for real-time robotic control. Experiments with a robotic manipulator tracking complex pose paths demonstrate the effectiveness of the approach. An open-source implementation is also provided.