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 , while recursive algorithms for the explicit calculation of the previously defined quantities in general tree-structured multibody systems are presented in .