Recursive linearized Newton-Euler formulation

With

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

$$\mbox{\boldmath$ Q $} = \left[\begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]$$ (A.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 $}}_0 = 0$.
  

  $$\delta \omega_i = \mbox{\boldmath$ R $}_i^T \{\delta \omega_{i-1} + \sigma_i [ \mbo... ..._i - \mbox{\boldmath$ Q $}(\omega_{i-1} + \dot{\theta}_i ) \delta \theta_i ] \}$$ (A.12)

  $$\delta \dot{\omega}_i = \mbox{\boldmath$ R $}_i^T \{ \delta \dot{\omega}_{i-1} + \sigma_i... ...ta}_i ) + \omega_{i-1} \times (\mbox{\boldmath$ z $}_0 \delta \dot{\theta}_i )]$$

  $$- \sigma_i \mbox{\boldmath$ Q $} [ \omega_{i-1} + \mbox{\boldmath... ...mega_{i-1} \times (\mbox{\boldmath$ z $}_0 \dot{\theta}_i )] \delta \theta_i \}$$ (A.13)

  $$\delta \dot{\mbox{\boldmath$ v $}}_i = \mbox{\boldmath$ R $}_i^T \{ \delta \dot{\mbox{\boldmath$ v $}}_{... ...sigma_i \mbox{\boldmath$ Q $} \dot{\mbox{\boldmath$ v $}}_{i-1} \delta \theta_i$$

  $$+ (1 -\sigma_i) [\mbox{\boldmath$ z $}_0 \delta \ddot{d}_i + 2 \d... ...{d}_i ) + 2 \omega_{i-1} \times (\mbox{\boldmath$ z $}_0 \delta \dot{d}_i )] \}$$

  $$+ \delta \dot{\omega}_i \times \mbox{\boldmath$ p $}_i + \delta \... ...th$ p $}_i) + \omega_i \times ( \delta \omega_i \times \mbox{\boldmath$ p $}_i)$$

  $$+ (1 - \sigma_i) (\dot{\omega}_i \times \mbox{\boldmath$ p $}_{di} + \omega_i \times ( \omega_i \times \mbox{\boldmath$ p $}_{di}) ) \delta d_i$$ (A.14)

  
  

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

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

  $$\delta \dot{\mbox{\boldmath$ v $}}_{ci} = \delta \mbox{\boldmath$ v $}_i + \delta \omega_i \times \mbox{\bo... ...ath$ r $}_i) + \omega_i \times (\delta \omega_i \times \mbox{\boldmath$ r $}_i)$$ (A.15)

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

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

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

  $$\delta \mbox{\boldmath$ n $}_i = \mbox{\boldmath$ R $}_{i+1} [ \delta \mbox{\boldmath$ n $}_{i+1} ... ...h$ f $}_{i} + \mbox{\boldmath$ r $}_{i} \times \delta \mbox{\boldmath$ F $}_{i}$$

  $$+ (1 - \sigma_i) (\mbox{\boldmath$ p $}_{di} \times \mbox{\boldma... ...\mbox{\boldmath$ R $}_{i+1} [ \mbox{\boldmath$ n $}_{i+1} ] \delta \theta_{i+1}$$ (A.19)

  $$\delta \tau_i = \sigma_i [ \delta \mbox{\boldmath$ n $}_i^T (\mbox{\boldmath$ R $... ...dmath$ R $}_i^T \mbox{\boldmath$ Q $} \mbox{\boldmath$ z $}_0) \delta \theta_i]$$

  $$+ (1 -\sigma_i) [ \delta \mbox{\boldmath$ f $}_i^T (\mbox{\boldmath$ R $}_i^T \mbox{\boldmath$ z $}_0) ]$$ (A.20)