Recursive linearized Newton-Euler formulation

[prev] [prev-tail] [tail] [up]

B.2 Recursive linearized Newton-Euler formulation

With

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

one can use the following

Forward Iterations for i = 1,2,…,n.
Initialize: δω₀ = δ₀ = δ₀ = 0.
$T T δωi = Ri δωi-1 + σi(z0δθ˙i - QR i ωiδθi) (B.12 ) δω˙i = RTi δ ˙wi-1 + σi[RTi δωi-1 × z0˙θi (B.13 ) T ˙ ¨ +R i ωi-1 × z0δθi + z0θi - (QRTi ˙ωi-1 + QRTi ωi- 1 × ωz0θ˙i)δθi] δv˙ = RT (δ˙ω × p + δω × (ω × p ) (B.14 ) i i i-1 i i-1 ) i-1 i +ωi-1 × (δωi- 1 × p )+ δv˙i ( i ) +(1 - σi)2δ ωi × z0˙di + 2ωi × z0δd˙i + z0δ¨di ( ) - σiQRTi ω˙i-1 × pi + ωi-1 × (wi-1 × pi)+ v˙i δθi T( ) +(1 - σi)R i ˙ωi- 1 × pdi + ωi-1 × (ωi-1 × pdi) δdi$
Backward Iterations for i = n,n - 1,…,1.
Initialize: δf_n+1 = δn_n+1 = 0.
$δ˙v = δ˙v + δ˙ω × r + δω × (ω × r ) (B.15 ) ci i i i i i i +ωi × (δωi × ri) δFi = miδ˙vci (B.16 ) δN i = Iciδω˙i + δωi × (Iciωi) + ωi × (Iciδωi) (B.17 ) δfi = Ri+1δf i+1 + δF i + σi+1Ri+1Qf i+1δθi+1 (B.18 ) δni = δN i + Ri+1 δni+1 + ri × δFi (B.19 ) +pi+1(× Ri+1δf i+1 ) +σi+1 Ri+1Qn + p × Ri+1Qf δθi+1 i+1 i+1 i+1 +(1 - σi+1)pdi+1pdi+1 × Ri+1f i+1δdi+1 δτi = σδnTi z0 + (1 - σi)δf Ti z0 (B.20 )$

[prev] [prev-tail] [front] [up]