This paper provides a strict system formulation for a class of nonholonomic systems with Lie bracket motions via rough path analysis. The dynamics of the resulting systems are represented by rough differential equations as augmented versions of ordinary differential equations. The rough differential equations are allowed to have rough signals generated by unbounded-variation functions and derived by classifying the functions according to "orders" of the boundedness. This paper clarifies rough

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... fferential equations driven by third-order rough signals, and the validity for control design problems is confirmed by considering a fourth-order chained system, which is a typical form of driftless affine nonholonomic systems.