Recursive linearized Newton-Euler formulation

With

$$\mbox{\boldmath$ p $}_{di} = \frac{\partial \mbox{\boldmath$ p $}_i}{\partial d_i} = \left[\begin{array}{ccc} 0 & -\sin\alpha_{i-1} & \cos\alpha_{i-1} \end{array}\right]^T$$ (B.10)

$$\mbox{\boldmath$ Q $} = \left[\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ (B.11)

one can use the following

• Forward Iterations for $i=1, 2, \ldots, n$.

  Initialize: $\delta \omega_0 = \delta \dot{\omega}_0 = \delta \dot{\mbox{\boldmath$ v $}}_0 = 0$.
  

  $$\delta \omega_i = \mbox{\boldmath$ R $}_i^T \delta \omega_{i-1} + \sigma_i (\mbox{\... ... \delta \dot{\theta}_i - \mbox{\boldmath$ QR $}_i^T \omega_i \delta \theta_i)$$ (B.12)

  $$\delta \dot{\omega}_i = \mbox{\boldmath$ R $}_i^T \delta \dot{\mbox{\boldmath$ w $}}_{i-1... ...ath$ R $}_i^T \delta \omega_{i-1} \times \mbox{\boldmath$ z $}_0 \dot{\theta}_i$$ (B.13)

  $$+ \mbox{\boldmath$ R $}_i^T \omega_{i-1} \times \mbox{\boldmath$ z $}_0 \delta \dot{\theta}_i + \mbox{\boldmath$ z $}_0 \ddot{\theta}_i$$

  $$- (\mbox{\boldmath$ QR $}_i^T \dot{\omega}_{i-1} + \mbox{\boldmath$ QR $}_i^T \omega_{i-1} \times \omega{z}_0 \dot{\theta}_i) \delta \theta_i ]$$

  $$\delta \dot{\mbox{\boldmath$ v $}}_i = \mbox{\boldmath$ R $}_i^T\Big(\delta \dot{\omega}_{i-1} \times \m... ... $}_i + \delta \omega_{i-1} \times(\omega_{i-1} \times \mbox{\boldmath$ p $}_i)$$ (B.14)

  $$+ \omega_{i-1} \times( \delta \omega_{i-1} \times \mbox{\boldmath$ p $}_i) + \delta \dot{\mbox{\boldmath$ v $}}_i\Big)$$

  $$+ (1-\sigma_i)\Big(2\delta \omega_i \times \mbox{\boldmath$ z $}_... ...h$ d $}}_i + \mbox{\boldmath$ z $}_0 \delta \ddot{\mbox{\boldmath$ d $}}_i\Big)$$

  $$- \sigma_i\mbox{\boldmath$ QR $}_i^T \Big(\dot{\omega}_{i-1} \tim... ... \times(w_{i-1}\times p_i) + \dot{\mbox{\boldmath$ v $}}_i\Big) \delta \theta_i$$

  $$+ (1-\sigma_i) \mbox{\boldmath$ R $}_i^T \Big(\dot{\omega}_{i-1} ... ...ega_{i-1}\times \mbox{\boldmath$ p $}_{di})\Big) \delta \mbox{\boldmath$ d $}_i$$

  
  

• Backward Iterations for $i=n, n-1, \ldots, 1$.

  Initialize: $\delta \mbox{\boldmath$ f $}_{n+1} = \delta \mbox{\boldmath$ n $}_{n+1} = 0$.
  

  $$\delta \dot{\mbox{\boldmath$ v $}}_{ci} = \delta \dot{\mbox{\boldmath$ v $}}_i + \delta \dot{\omega}_i\time... ...dmath$ r $}_i + \delta \omega_i \times(\omega_i \times \mbox{\boldmath$ r $}_i)$$ (B.15)

  $$+ \omega_i \times ( \delta \omega_i \times \mbox{\boldmath$ r $}_i)$$

  $$\delta \mbox{\boldmath$ F $}_i = m_i \delta \dot{\mbox{\boldmath$ v $}}_{ci}$$ (B.16)

  $$\delta \mbox{\boldmath$ N $}_i = \mbox{\boldmath$ I $}_{ci} \delta \dot{\omega}_i + \delta \omega_... ...}_{ci} \omega_i) + \omega_i \times (\mbox{\boldmath$ I $}_{ci} \delta \omega_i)$$ (B.17)

  $$\delta \mbox{\boldmath$ f $}_i = \mbox{\boldmath$ R $}_{i+1} \delta \mbox{\boldmath$ f $}_{i+1} + ... ...1} \mbox{\boldmath$ R $}_{i+1} \mbox{\boldmath$ Qf $}_{i+1} \delta \theta_{i+1}$$ (B.18)

  $$\delta \mbox{\boldmath$ n $}_i = \delta \mbox{\boldmath$ N $}_i + \mbox{\boldmath$ R $}_{i+1} \del... ...math$ n $}_{i+1} + \mbox{\boldmath$ r $}_i\times \delta \mbox{\boldmath$ F $}_i$$ (B.19)

  $$+ \mbox{\boldmath$ p $}_{i+1} \times \mbox{\boldmath$ R $}_{i+1} \delta \mbox{\boldmath$ f $}_{i+1}$$

  $$+ \sigma_{i+1} \Big( \mbox{\boldmath$ R $}_{i+1} \mbox{\boldmath$... ...ox{\boldmath$ R $}_{i+1} \mbox{\boldmath$ Qf $}_{i+1} \Big) \delta \theta_{i+1}$$

  $$+ (1-\sigma_{i+1} ) \mbox{\boldmath$ p $}_{di+1} \mbox{\boldmath$... ...} \times \mbox{\boldmath$ R $}_{i+1} \mbox{\boldmath$ f $}_{i+1} \delta d_{i+1}$$

  $$\delta \mbox{\boldmath$ \tau $}_i = \sigma \delta\mbox{\boldmath$ n $}_i^T \mbox{\boldmath$ z $}_0 + (1 - \sigma_i) \delta\mbox{\boldmath$ f $}_i^T \mbox{\boldmath$ z $}_0$$ (B.20)