We give now a brief summary of the Contact and
Collision Algorithms.
If the tip contact constraint (free foot touching the ground) is given
holonomically as 
, then by taking time derivatives
we also obtain
where 
.
Multiplying (
) by 
and substituting for 
using (), one obtains an operator expression for 
.
is
a square matrix of dimension equal to the number of constraints, and it is
a quantity related to the Khatib operational space inertia.
are the free generalized accelerations without the influence of the
contact force in the dynamics.
The final expression for is expressed in terms of the constrained
components of the spatial acceleration 
,
where 
.
The quantity 
likewise is composed of
the constrained components of the linear and angular
velocities for the various links in the multibody system.
The true angle accelerations are the sum of and a
correction term 
which results from the contact forces
propagating throughout the body.
These correction accelerations can be calculated from by the
relationship
A very similar algorithm exists for calculating the change in velocities
due to an inelastic collision with the ground.
The change in the generalized velocities will depend on the leg
tip velocities at the moment of contact with the ground, 
.
One solves for the impulse force 
,
One may solve for 
in
to obtain the generalized velocities after collision 
.
The Contact and Collision Algorithms are discussed
at greater length in [1], while
recursive algorithms for the explicit calculation of the previously
defined quantities in general tree-structured multibody systems are
presented in [6].
