Consider the control system (E) given by x = x(f + ug) where x∈SO(3), |u|≤1 and f, g∈so(3) define two perpendicular left-invariant vector fields normalized so that ||f|| = cos(α) and ||g|| = sin(α), α∈(0, π/4). In this paper, we provide an upper bound and a lower bound for N(α), the maximum number of switchings for time-optimal trajectories of (E). More precisely, we show that N{sub}s (α) ≤N(α) ≤N{sub}s (α) + 4, where N{sub}s (α) is a suitable integer function of a such that N{sub}s (α) ～π/4α(α→0). The result is obtained by studying the time optimal synthesis of a projected control problem on RP{sup}2, where the projection is defined by an appropriate Hopf fibration.
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