Recursive Newton-Euler formulation

Forward Iterations for i = 1,2,…,n.
Initialize: ω₀ = ₀ = 0 and ₀ = -g.
$T ωi = Ri [ωi-1 + σiz0˙θi] (A.1 ) ω˙i = RTi {˙ωi-1 + σi[z0¨θi + ωi-1 × (z0 ˙θi)]} (A.2 ) T ¨ ˙ v˙i = Ri {˙vi-1 + (1 - σi)[z0di + 2ωi- 1 × (z0di)]} +ω˙i × pi + ωi × (ωi × pi) (A.3 )$
Backward Iterations for i = n,n - 1,…,1.
Initialize: f_n+1 = n_n+1 = 0.
$˙vci = ˙vi + ˙ωi × ri + ωi × (ωi × ri) (A.4 ) F = m ˙v (A.5 ) i i ci N i = Iciω˙i + ωi × (Iciωi) (A.6 ) fi = Ri+1[fi+1]+ F i (A.7 ) ni = Ri+1[ni+1]+ pi × f i + N i + ri × F i (A.8 ) τi = σinTi (RTi z0)+ (1 - σi)fTi (RTi z0) (A.9 )$