Time optimal synthesis for a so(3)-left-invariant control system on a sphere

Abstract

Consider the control system (/spl Sigma/) given by x/spl dot/ = x(f + ug) where x /spl isin/ SO(3), |u| /spl les/ 1 and f, g /spl isin/ so(3) define two perpendicular left-invariant vector fields normalized so that ||f|| = cos(/spl alpha/) and ||g|| = sin(/spl alpha/), /spl alpha/ /spl isin/ (0, /spl pi//4). In this paper, we provide an upper bound and a lower bound for N(/spl alpha/), the maximum number of switchings for time-optimal trajectories of (/spl Sigma/). More precisely, we show that N/sub S/(/spl alpha/) /spl les/ N(/spl alpha/) /spl les/ N/sub S/(/spl alpha/) /spl les/+4, where N/sub S/(a) is a suitable integer function of /spl alpha/ such that N/sub S/(/spl alpha/) /sub /spl alpha//spl rarr/0//sup /spl sim//. The result is obtained by studying the time optimal synthesis of a projected control problem on /spl Ropf/P/sup 2/, where the projection is defined by an appropriate Hopf fibration.