Template Class ActionModelNumDiffTpl

Defined in File action.hpp

Inheritance Relationships

Base Type

public crocoddyl::ActionModelAbstractTpl< _Scalar > (Template Class ActionModelAbstractTpl)

Class Documentation

template<typename _Scalar> class ActionModelNumDiffTpl : public crocoddyl::ActionModelAbstractTpl<_Scalar>

This class computes the numerical differentiation of an action model.

It computes the Jacobian of the cost, its residual and dynamics via numerical differentiation. It considers that the action model owns a cost residual and the cost is the square of this residual, i.e., \(\ell(\mathbf{x},\mathbf{u})=\frac{1}{2}\|\mathbf{r}(\mathbf{x},\mathbf{u})\|^2\), where \(\mathbf{r}(\mathbf{x},\mathbf{u})\) is the residual vector. The Hessian is computed only through the Gauss-Newton approximation, i.e.,

\[\begin{split}\begin{eqnarray*} \mathbf{\ell}_\mathbf{xx} &=& \mathbf{R_x}^T\mathbf{R_x} \\ \mathbf{\ell}_\mathbf{uu} &=& \mathbf{R_u}^T\mathbf{R_u} \\ \mathbf{\ell}_\mathbf{xu} &=& \mathbf{R_x}^T\mathbf{R_u} \end{eqnarray*}\end{split}\]

where the Jacobians of the cost residuals are denoted by \(\mathbf{R_x}\) and \(\mathbf{R_u}\). Note that this approximation ignores the tensor products (e.g., \(\mathbf{R_{xx}}\mathbf{r}\)).

Finally, in the case that the cost does not have a residual, we set the Hessian to zero, i.e., \(\mathbf{L_{xx}} = \mathbf{L_{xu}} = \mathbf{L_{uu}} = \mathbf{0}\).