As explained in class dynamicgraph::sot::FeatureAbstract,

\[ {\bf E}(t) = {\bf e}({\bf q}(t), t)= {\bf s}({\bf q}(t)) - {\bf s}^*(t) \]

thus:

\[ \dot{\bf E}= \frac{\partial{\bf e}}{\partial{\bf q}} \dot{\bf q} + \frac{\partial{\bf e}}{\partial t} \\ \]

The features are responsible for computing:

\( {\bf E}(t) \)
\( \frac{\partial{\bf e}}{\partial t} \)
\( \frac{\partial{\bf e}}{\partial{\bf q}} \)

The class dynamicgraph::sot::Task takes some features outputs as inputs. It imposes an exponential decrease, i.e. \( \dot{\bf E} = -\lambda {\bf E} \). It gives:

\[ -\lambda {\bf e} = \frac{\partial{\bf e}}{\partial{\bf q}} \dot{\bf q} + \frac{\partial{\bf e}}{\partial t} \]

and computes solutions as:

\[ \dot{\bf q} = \frac{\partial{\bf e}}{\partial{\bf q}}^{\dagger} \left( - \lambda {\bf e} - \frac{\partial{\bf e}}{\partial t} \right) + K v \]

where:

\(\frac{\partial{\bf e}}{\partial{\bf q}}^{\dagger}\) is the pseudo-inverse of \(\frac{\partial{\bf e}}{\partial{\bf q}}\),
\(\dot{\bf q} \in \mathbb{R}^n\),
\(K \in \mathcal{M}_{n,m}(\mathbb{R})\) is a base of the kernel of \(\frac{\partial{\bf e}}{\partial{\bf q}}\), of dimension \(m\),
and \(v\) spans \(\mathbb{R}^m\) so that \(K v\) spans the kernel of \(\frac{\partial{\bf e}}{\partial{\bf q}}\).

\(v\) is a free parameters left to tasks of lower priority.