Recursive Newton-Euler formulation

Forward Iterations for i = 1,2,…,n.
Initialize: ω₀ = ₀ = 0 and ₀ = -g.
$T ωi = Ri ωi-1 + σiz0θ˙i (B.1 ) ω˙i = RTi ˙ωi-1 + σiRTi ωi-1 × z0˙θi + σiz0¨θi (B.2 ) T v˙i = R i (ω˙i-1 × pi + ωi- 1 × (ωi-1 × pi) + ˙vi-1) (B.3 ) +(1 - σi)(2ωi × z0d˙i + z0d¨i)$
Backward Iterations for i = n,n - 1,…,1.
Initialize: f_n+1 = n_n+1 = 0.
$˙vci = ˙ωi × ri + ωi × (ωi × ri)+ v˙i (B.4 ) F = m ˙v (B.5 ) i i ci N i = Ici¨ωi + ωi × Iciωi (B.6 ) fi = Ri+1f i+1 + F i (B.7 ) ni = N i + Ri+1ni+1 + ri × Fi + pi+1 × Ri+1f i+1 (B.8 ) τi = σiniz ) + (1- σi)f Ti z0 (B.9 )$