Applying actions

Up to this point, it has been assumed that no actions are applied to the mechanical system. This is the way the Euler-Lagrange equation usually appears in physics because the goal is to predict motion, rather than control it. Let $ u \in {\mathbb{R}}^n$ denote an action vector. Actions can be applied to the Lagrangian formulation as generalized forces that ``act'' on the right side of the Euler-Lagrange equation. This results in

$\displaystyle \frac{d}{dt} \frac{\partial L}{\partial {\dot q}} - \frac{\partial L}{\partial q} = u .$ (13.136)

The actions force the mechanical system to deviate from its usual behavior. In some instances, the true actions may be expressed in terms of other variables, and then $ u$ is obtained by a transformation (recall transforming action variables for the differential drive vehicle of Section 13.1.2). In this case, $ u$ may be replaced in (13.136) by $ \phi(u)$ for some transformation $ \phi $. In this case, the dimension of $ u$ need not be $ n$.

Steven M LaValle 2012-04-20