There are several constraints which must be enforced along the duration of the walking step. First, we must ensure that the swing leg always remain above ground while in motion during phase 1. This involves a series of simple kinematic calculations. Additionally, we require that the vertical component of the contact forces in each phase remain positive so that there is not a premature liftoff of the leg from the ground. If this component does not remain positive and we are still constraining the leg to be in contact, then the configuration will not make sense.
These contact constraints do require a significant amount of computation in order to evaluate them, and we have determined them to be necessary as they can easily be violated if they are left out. In fact, the necessity of these constraints inspired us to begin working with a more complex tree-structured model with the torso as the base of the biped and full positional degrees of freedom in the plane. A simpler model would take the foot which is always in contact with the ground during one step to be the base. Several investigators take this approach in their handling of the problem, but this prevents the verification of the contact force inequalities.