Clarke Coordinates are Generalized Improved State Parametrization for Continuum Robots

In this letter, we demonstrate that previously proposed improved state parameterizations for soft and continuum robots are specific cases of Clarke coordinates. By explicitly deriving these improved parameterizations from a generalized Clarke transformation matrix, we unify various approaches into one comprehensive mathematical framework. This unified representation provides clarity regarding their relationships and generalizes them beyond existing constraints, including arbitrary numbers of joints, joint distributions, and underlying modeling assumptions. This unification consolidates prior insights and establishes Clarke coordinates as a foundational tool, enabling systematic knowledge transfer across different subfields within soft and continuum robotics.