Define the Lagrangian (Fossen, 2002),
where X e Rr is a vector of Lagrange multipliers. Consequently, differentiating the Lagrangian L with respect to f, yields
Next, assume that TW-1TT is non-singular such that t = Tf = 1 TW-1TtX ^ X = 2(TW-1Tt )-1 t.
Substituting (1.28) into (1.27) yields, f = T* t, T* = W-1Tt (TW-1Tt
where T* is recognized as the generalized inverse. For the case W=I, that is equally weighted control forces, (1.29) reduces to the Moore-Penrose pseudo inverse,
Since f = T*t, the control input vector u can be computed from (1.7) as,
Notice that this solution is valid for all a but not optimal with respect to a time-varying a.
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