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 . 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, .