The state equations of the biped walker are those of a multibody system experiencing contact forces,
In equation (
),
is the square, positive-definite mass-inertia matrix,
is the vector of Coriolis and centrifugal forces,
is a vector of gravitational forces,
u are the applied torques at the links, 
is the constraint Jacobian,
and 
is the constraint force.
Several different approaches to recursive, symbolic multibody algorithms
were studied, compared, and represented in a unifying
formalism in [6].
This work also included the extension of several algorithms to multiple
degree of freedom joints and tree-structured systems.
The approach is based on decomposing the dynamical quantities
into physical, matrix operators.
The various link operations are stacked into larger, matrix operators which,
in turn, provide a very clean notation which can easily be manipulated for
estimation and control design purposes.
For high dimensions, recursive, symbolic
dynamical models are more efficient for calculating the forward dynamics
than other non-recursive procedures which require constructing and inverting
the entire mass-inertia matrix, .
