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This paper proposes a switching control strategy for a class of symmetric affine systems whose accessibility Lie algebras are spanned by the input vector fields and their first order Lie brackets. The key idea here is to decompose a given system into several (chained) subsystems, and then to apply a conventional control method for these subsystems in sequence. As for the conventional part, we adopt the time-state control method. The proposed strategy guarantees convergence of the state to an arbitrarily given neighborhood of the origin. Finally, with a numerical example, we show that a new type of mobile robot which can be approximated by a system in the class applied to the strategy is valid as well.