Recursive linearized Newton-Euler formulation

With

∂pi [ ]T pdi = -∂d-= 0 sinαi cosαi (A.10 ) ⌊ i ⌋ | 0 - 1 0 | Q = ⌈ 1 0 0 ⌉ (A.11 ) 0 0 0

one can use the following

Forward Iterations for i = 1,2,…,n.
Initialize: δω₀ = δ₀ = δ₀ = 0.
$δωi = RTi {δωi-1 + σi[z0δ˙θi - Q (ωi-1 + θ˙i)δθi]} (A.12) δω˙i = RT {δ˙ωi-1 + σi[z0δ¨θi + δωi-1 × (z0 ˙θi) + ωi- 1 × (z0δθ˙i)] i ¨ ˙ - σiQ [ωi- 1 + z0θi + ωi-1 × (z0θi)]δθi} (A.13) δv˙i = RTi {δ˙vi-1 - σiQ ˙vi-1δθi +(1 - σ )[z δ¨d + 2δω × (z d˙)+ 2ω × (z δ ˙d)]} i 0 i i-1 0 i i-1 0 i +δω˙i × pi + δωi × (ωi × pi)+ ωi × (δωi × pi) +(1 - σi)(ω˙i × pdi + ωi × (ωi × pdi))δdi (A.14)$
Backward Iterations for i = n,n - 1,…,1.
Initialize: δf_n+1 = δn_n+1 = 0.
$δ˙vci = δ˙vi + δ˙ωi × ri + δωi × (ωi × ri)+ ωi × (δωi × ri)(A.15 ) δFi = miδ˙vci (A.16 ) δN = I δω˙ + δω × (I ω) + ω × (I δω ) (A.17 ) i ci i i ci i i ci i δfi = Ri+1[δfi+1]+ δF i + σi+1QRi+1 [fi+1]δθi+1 (A.18 ) δni = Ri+1[δni+1]+ δN i + pi × δfi + ri × δF i +(1 - σi)(pdi × f i)δdi + σi+1QRi+1 [ni+1 ]δθi+1(A.19 ) δτi = σi[δnTi (RTi z0)- nTi (RTi Qz0 )δθi] +(1 - σi)[δfT (RT z0 )] (A.20 ) i i$