The following general procedure can be followed to derive the
differential model using Lagrangian mechanics on a coordinate
neighborhood of a smooth -dimensional manifold:
- Determine the degrees of freedom of the system and define the
appropriate -dimensional smooth manifold .
- Express the kinetic energy as a quadratic form in the
configuration velocity components:
- Express the potential energy .
be the Lagrangian function,
and use the Euler-Lagrange equation (13.130) to
determine the differential constraints.
- Convert to phase space form by letting
possible, solve for to obtain
(2D Rigid Body Revisited)
The equations in (13.109) can be alternatively derived
using the Euler-Lagrange equation. Let
, and let
to conform to the
notation used to express the Lagrangian.
The kinetic energy is the sum of kinetic energies due to linear and
angular velocities, respectively. This yields
are the mass and moment of inertia, respectively.
Assume there is no gravity; hence,
Suppose that generalized forces , , and can be
applied to the configuration variables. Applying the Euler-Lagrange
These expressions are equivalent to those given in
). One difference is that conversion to the
phase space is needed. The second difference is that the action
variables in (13.139
) do not refer directly to forces
and moments. They are instead interpreted as generalized forces
act on the configuration variables. A conversion should be performed
if the original actions in (13.109
) are required.
Steven M LaValle