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Geodesic Least Mean Square algorithm for 3D tracking
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DOI：10.1016/j.dsp.2026.106210.png)
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Abstract
En 中文
Quaternions are well-known for modelling three-dimensional rotations. As such, they have been considered in tracking three-dimensional objects in orientation applications which require modelling in a curved manifold (e.g. on the surface of a sphere). However, quaternion-valued learning algorithms such as the Quaternion Least Mean Square algorithm do not cater for the curved manifold. To address this shortcoming in the literature, we propose the Geodesic Least Mean Square algorithm. Its novelty stems from two contributions. First, the cost function is based on the quaternion geodesic distance rather than the traditional formulation of the error as an arithmetic difference. Second, its weight is modelled as 3D rotation, thus making it an interpretable artificial intelligence algorithm. Simulation studies on both synthetic and real-world data support the approach.
Keywords:
quaternions
interpretable AI
least mean square
geodesic distance
3D orientations
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