Special recursive algorithms have also been developed to determine the contact forces experienced by the biped under contact and the resulting generalized accelerations which are produced. See Appendix , also see [7] and the references found therein.

An important component of our dynamical modeling which also contributed to our ability to solve this problem was the development of a Reduced Dynamics Algorithm which makes use of the Contact Algorithm and which is more fully described in Appendix . Because of the contact constraints, we are faced with a differential-algebraic system. Two courses of actions are possible when it is necessary to integrate the dynamics, one being the use of specially tailored integration routines which often require the partial derivatives of the various contact constraints. The preferable approach, however, is to use a reduced unconstrained set of dynamics which evolve on the constraint manifold. Then it is possible to use standard integration procedures. This latter approach is the one we take.

In the first phase of the biped motion, where one leg is swinging, the contact constraints reduce the total degrees of freedom from 7 to 5. Thus, using the Reduced Dynamics Algorithm, an unconstrained 10-dimensional state space can represent the system during this period instead of the full 14 dimensions for the completely free system. The remaining 4 states and their time derivatives can be determined from the 10 independent states and their time derivatives. Similarly in the shorter second phase, when both feet are in contact with the ground, contact constraints allows us to work with a system with only a 6-dimensional state space.

Mon Oct 11 17:19:43 MET DST 1999