Template Class ActionModelAbstractTpl

Defined in File action-base.hpp

Inheritance Relationships

Base Type

public crocoddyl::ActionModelBase (Class ActionModelBase)

Derived Types

public crocoddyl::ActionModelImpulseFwdDynamicsTpl< _Scalar > (Template Class ActionModelImpulseFwdDynamicsTpl)
public crocoddyl::ActionModelLQRTpl< _Scalar > (Template Class ActionModelLQRTpl)
public crocoddyl::ActionModelNumDiffTpl< _Scalar > (Template Class ActionModelNumDiffTpl)
public crocoddyl::ActionModelUnicycleTpl< _Scalar > (Template Class ActionModelUnicycleTpl)
public crocoddyl::IntegratedActionModelAbstractTpl< _Scalar > (Template Class IntegratedActionModelAbstractTpl)

Class Documentation

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

Abstract class for action model.

An action model combines dynamics, cost functions and constraints. Each node, in our optimal control problem, is described through an action model. Every time that we want describe a problem, we need to provide ways of computing the dynamics, cost functions, constraints and their derivatives. All these is described inside the action model.

Concretely speaking, the action model describes a time-discrete action model with a first-order ODE along a cost function, i.e.

the state \(\mathbf{z}\in\mathcal{Z}\) lies in a manifold described with a nx-tuple,
the state rate \(\mathbf{\dot{x}}\in T_{\mathbf{q}}\mathcal{Q}\) is the tangent vector to the state manifold with ndx dimension,
the control input \(\mathbf{u}\in\mathbb{R}^{nu}\) is an Euclidean vector
\(\mathbf{r}(\cdot)\) and \(a(\cdot)\) are the residual and activation functions (see ResidualModelAbstractTpl and ActivationModelAbstractTpl, respetively),
\(\mathbf{g}(\cdot)\in\mathbb{R}^{ng}\) and \(\mathbf{h}(\cdot)\in\mathbb{R}^{nh}\) are the inequality and equality vector functions, respectively.

The computation of these equations are carried out out inside calc() function. In short, this function computes the system acceleration, cost and constraints values (also called constraints violations). This procedure is equivalent to running a forward pass of the action model.

However, during numerical optimization, we also need to run backward passes of the action model. These calculations are performed by calcDiff(). In short, this method builds a linear-quadratic approximation of the action model, i.e.:

\[\begin{split} \begin{aligned} &\delta\mathbf{x}_{k+1} = \mathbf{f_x}\delta\mathbf{x}_k+\mathbf{f_u}\delta\mathbf{u}_k, &\textrm{(dynamics)}\\ &\ell(\delta\mathbf{x}_k,\delta\mathbf{u}_k) = \begin{bmatrix}1 \\ \delta\mathbf{x}_k \\ \delta\mathbf{u}_k\end{bmatrix}^T \begin{bmatrix}0 & \mathbf{\ell_x}^T & \mathbf{\ell_u}^T \\ \mathbf{\ell_x} & \mathbf{\ell_{xx}} & \mathbf{\ell_{ux}}^T \\ \mathbf{\ell_u} & \mathbf{\ell_{ux}} & \mathbf{\ell_{uu}}\end{bmatrix} \begin{bmatrix}1 \\ \delta\mathbf{x}_k \\ \delta\mathbf{u}_k\end{bmatrix}, &\textrm{(cost)}\\ &\mathbf{g}(\delta\mathbf{x}_k,\delta\mathbf{u}_k)<\mathbf{0}, &\textrm{(inequality constraint)}\\ &\mathbf{h}(\delta\mathbf{x}_k,\delta\mathbf{u}_k)=\mathbf{0}, &\textrm{(equality constraint)} \end{aligned} \end{split}\]

where

\(\mathbf{f_x}\in\mathbb{R}^{ndx\times ndx}\) and \(\mathbf{f_u}\in\mathbb{R}^{ndx\times nu}\) are the Jacobians of the dynamics,
\(\mathbf{\ell_x}\in\mathbb{R}^{ndx}\) and \(\mathbf{\ell_u}\in\mathbb{R}^{nu}\) are the Jacobians of the cost function,
\(\mathbf{\ell_{xx}}\in\mathbb{R}^{ndx\times ndx}\), \(\mathbf{\ell_{xu}}\in\mathbb{R}^{ndx\times nu}\) and \(\mathbf{\ell_{uu}}\in\mathbb{R}^{nu\times nu}\) are the Hessians of the cost function,
\(\mathbf{g_x}\in\mathbb{R}^{ng\times ndx}\) and \(\mathbf{g_u}\in\mathbb{R}^{ng\times nu}\) are the Jacobians of the inequality constraints, and
\(\mathbf{h_x}\in\mathbb{R}^{nh\times ndx}\) and \(\mathbf{h_u}\in\mathbb{R}^{nh\times nu}\) are the Jacobians of the equality constraints.

Additionally, it is important to note that calcDiff() computes the derivatives using the latest stored values by calc(). Thus, we need to first run calc().