A new concept of a dynamical system representation is proposed. The idea of the representation is simple: combination of nonlinear coordinate transformation and time scale transformation. Introducing a projected referential trajectory and a moving frame on it, the coordinate transformation separates a system state into sub-states tangent to the trajectory and orthogonal one. Then time scale transformation using the arc length of the trajectory yields a state equation in which the state evolves spatially around the trajectory. As one of the advantages of the proposed method, time scale transformation enables us to ignore dynamics of the tangent sub-state at the same time. Therefore, the proposed representation realizes one-degree model reduction for smooth control systems. Since all transformation used in this article is invertible, this reduction is recoverable.
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